meds-c 2015 - sfu.ca · 2016-10-04 · meds-c 2015 was organized by members of the mathematics...

SIMON FRASER UNIVERITY | FACULTY OF EDUCATION

MEDS-C 2015 PROCEEDINGS OF THE 10th ANNUAL MATHEMATICS

EDUCATION DOCTORAL STUDENTS CONFERENCE

OCTOBER 17, 2015

MEDS-C 2015 Programme

TABLE OF CONTENTS

MEDS-C 2015 PROGRAMME – OCTOBER 17, 2015 ...................................................................................... 1

CONTRIBUTIONS ........................................................................................................................................................ 2

PLENARY SPEAKER ................................................................................................................................................... 3

ABSTRACTS .................................................................................................................................................................. 4

TEACHERS TEACH, STUDENTS LEARN? ........................................................................................................... 9

DARIEN ALLAN

AN EXPLORATION INTO A SAUDI MATHEMAICS TEACHER’S FIGURED WORLDS ....................... 17

LYLA ALSALIM

A PHENOMENOLOGY PERSPECTIVE TO STUDY PROFESSIONAL DEVELOPMENT ....................... 24

MELANIA ALVAREZ

“WHAT WE NEED TO SHOW IS THAT T IS WELL-DEFINED”: GESTURE AND DIAGRAM IN

ABSTRACT ALGEBRA.............................................................................................................................................. 32

ANDREW HARE

NEGOTIATING MEANING: A CASE OF TEACHERS DISCUSSING MATHEMATICAL

ABSTRACTION IN THE BLOGOSPHERE........................................................................................................... 40

JUDY LARSEN

DISCURSIVE PRACTICES USED IN DEFINING MATHEMATICAL LEARNING DISABILITIES: A

BRIEF TEXTBOOK ANALYSIS .............................................................................................................................. 48

PETER LEE

STUDENTS’ MODELLING PROCESS – A CASE STUDY ................................................................................ 56

MINNIE LIU

UNDERGRADUATE STUDENTS’ PERCEPTION OF TRANSFORMATION OF SINUSOIDAL

FUNCTIONS ................................................................................................................................................................ 64

MASOMEH JAMSHID NEJAD

COHERENT AND DIVERGING DISCOURSE IN MATHEMATICAL ACTIVITIES WITH DYNAMIC

GEOMETRY ................................................................................................................................................................. 71

OI-LAM NG

IDENTITY AND MATHEMATIS: A PRELIMINARY EXPLORATION ........................................................ 79

TANYA NOBLE

STUDENTS’ FLOW EXPERIENCES IN THREE TYPES OF MATHEMATICAL PROBLEMS AND

TWO DIFFERENT EDUCATIONAL ENVIRONMENTS .................................................................................. 84

ZAKIEH PARHIZGAR, HASSAN ALAMOLHODAEI

“THE OTHER TEN”: AUDEN’S SENSE OF NUMBER ..................................................................................... 92

SHEREE RODNEY

TEACHER TENSIONS: THE CASE OF NAOMI .............................................................................................. 100

ANNETTE ROULEAU


A BLIND UNDERGRADUATE STUDENT’S JOURNEY TOWARD AN UNDERSTANDING OF

PRE-CALCULUS CONCEPTS ............................................................................................................................... 106

MINA SEDAGHATJOU

MATHEMATICS LEARNING AMONG UNDERGRADUATES ON THE AUTISM SPECTRUM ........ 114

JEFFREY TRUMAN

OVERVIEW OF RESEARCH ON STUDENTS’ VIEWS OF ORAL ASSESSMENT IN MATHEMATICS

...................................................................................................................................................................................... 122

MILICA VIDENOVIC


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MEDS-C 2015 PROGRAMME – OCTOBER 17, 2015

08:30 – 09:00 Welcome and Coffee

EDB 7608 EDB 7610

09:00 – 09:35 Judy Larsen

Negotiating Meaning: A Case of Teachers

Discussing Mathematical Abstraction in the

Blogosphere

Mina SedaghatJou

A Novel Approach on Enabling Advanced

Mathematical Communication in Absence of Sight

09:40 – 10:15 Tanya Noble

Identity and Mathematics: A Preliminary

Exploration

Peter Lee

Discursive Practices Used in Defining Mathematical

Learning Disabilities: A Brief Textbook Analysis

10:15 – 10:25 Break

10:25 – 11:00 Zakieh Parhizgar

Students’ Flow Experiences in Three Types

of Mathematical Problems and Two

Different Educational Environments

Minnie Liu

Students’ Modelling Process with a Focus on

Mathematization – A Case Study

11:00 – 11:45 Plenary Speaker: Cynthia Nicol

Slow Pedagogy, Research and Relations:

Building relationships for research that matter

11:45 – 12:00 Plenary Q & A

12:00 – 13:15 Lunch

13:15 – 13:50 Masomeh Jamshid Nejad

Undergraduate Students’ Perception of

Transformation of Sinusoidal Functions

Darien Allan

Teachers Teach, Students Learn?

13:55 - 14:30 Annette Rouleau

Teacher Tensions: The Case of Naomi

Sheree Gillings Rodney

The Other Ten: A Case of Auden

14:30 – 14:40 Break

14:40 – 15:15 Melania Alvarez

A Phenomenology Perspective to Study

Professional Development

Jeffrey Truman

Mathematics Learning Among Undergraduates on

the Autism Spectrum

15:15 – 15:50 Andrew Hare

“What we need to show is that T is

well-defined”: Gesture and Diagram in

Abstract Algebra

Milica Videnovic

Overview of Research on Students’ Views of Oral

Assessment in Mathematics

15:50 – 16:00 Break

16:00 – 16:35 Oi-Lam Ng

Commensurability of Discourse in Mathematical Activities with Dynamic Geometry

16:40 – 17:00 Wrap up

MEDS-C 2015 Contributions

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CONTRIBUTIONS

MEDS-C 2015 was organized by members of the Mathematics Education Doctoral

Program. The conference would not be possible without the following contributions:

Conference Coordinators: Darien Allen and Minnie Liu

Program Coordinators: Lyla Alsalim and Tanya Noble

Review Coordinators: Sheree Rodney and Mina Sedaghatjou

Proceedings Editors: Peter Lee, Andrew Hare, and Judy Larsen

Technology Support Team: Oi-Lam Ng and Mina Sedaghatjou

Timer: Jeffrey Truman

Lunch Coordinator: Melania Alvarez

Snack Coordinators: Milica Videnovic and Annette Rouleau

MEDS-C 2015 Contributions

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PLENARY SPEAKER

CYNTHIA NICOL

SLOW PEDAGOGY, RESEARCH AND RELATIONS:

BUILDING RELATIONSHIPS FOR RESEARCH THAT MATTER

This presentation explores mathematics education research that interrupts, intervenes

and is intentionally participatory. I consider how theories of place and land-based

participatory research methodologies can come together to better understand

culturally responsive mathematics education. At the core of any participatory, activist

research is the need for relationship building. Drawing upon examples from various

projects I consider the concept of slow pedagogy and research as a way of building

relationships for research that ultimately makes a difference in the world.

MEDS-C 2015 Abstracts

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ABSTRACTS

DARIEN ALLAN

TEACHERS TEACH, STUDENTS LEARN?

This paper approaches the issue of student learning from the student perspective.

Leontiev’s activity theory is used to analyse observed student behaviours and student

goals to deduce student motive. In this study three student amalgams are analysed in

the particular activity setting of taking notes. Although the observed actions are

similar, the results show that the primary motives are different, and that this ultimately

had consequences for their learning. Thus, what teachers notice in the classroom may

not provide enough information to determine whether or not learning will occur.

LYLA ALSALIM

AN EXPLORATION INTO A SAUDI MATHEMAICS TEACHER’S FIGURED

WORLDS

In this paper, patterns-of-participation theory serves as a lens to interpret and

understand Saudi high school mathematics teachers’ practices. This framework

focuses mainly on understanding what practices and figured worlds are significant for

the teacher and how the teacher engages in those figured worlds. The data presented is

about Abeer, a high school mathematics teacher in Saudi Arabia. The data generated

suggests that there are five significant practices or figured worlds to Abeer’s sense of

her practice as a mathematics teacher. The paper discusses and explains these figured

worlds.

MELANIA ALVAREZ

A PHENOMENOLOGY PERSPECTIVE TO STUDY PROFESSIONAL

DEVELOPMENT

The purpose of this phenomenological study is to provide a phenomenological

framework to study professional development.

ANDREW HARE

“WHAT WE NEED TO SHOW IS THAT T IS WELL-DEFINED”: GESTURE AND

DIAGRAM IN ABSTRACT ALGEBRA

Proving that a mapping is well-defined is a key step in many results in group theory.

Such an argument is required in the Orbit-Stabilizer Theorem, as well as the theorem

that gives a necessary and sufficient condition for the set of cosets of a subgroup to

form a group (namely, that the subgroup is a normal subgroup). It can be challenging

for students to even realize that such an argument must be provided, as it is the first


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time they have been exposed to functions defined on equivalence classes. This paper

studies the gestures and diagrams that one professor uses to explain the concept of

well-definedness.

JUDY LARSEN

NEGOTIATING MEANING: A CASE OF TEACHERS DISCUSSING

MATHEMATICAL ABSTRACTION IN THE BLOGOSPHERE

Many mathematics teachers engage in the practice of blogging. Although they are

separated geographically, they are able to discuss teaching-related issues. In an effort

to better understand the nature of these discussions, this paper presents an analysis of

one particular episode of such a discussion. Wenger’s theoretical framework of

communities of practice informs the analysis by providing a tool to explain the

negotiation of meaning in the episode. Results indicate that the blogging medium

supports continuity of discussions and can allow for the negotiation of meaning, but

that a more nuanced treatment of the construct is necessary.

PETER LEE

DISCURSIVE PRACTICES USED IN DEFINING MATHEMATICAL LEARNING

DISABILITIES: A BRIEF TEXTBOOK ANALYSIS

This paper compares two separate descriptions of mathematical learning disabilities

given in a textbook on learning disabilities. One description is from an early edition of

the textbook and the other description is from a later edition of the same textbook.

Critical discourse analysis is applied to both descriptions to investigate the changing

discursive practices used in defining mathematical learning disabilities. While a

“medical discourse” dominates both the early and later editions, there is evidence of

other competing discourses which suggests some instability around the boundaries of

what constitutes a “mathematical learning disability.”

MINNIE LIU

STUDENTS’ MODELLING PROCESS – A CASE STUDY

This article presents a case study in which three grade 8 students worked

collaboratively to solve a modelling task. Data indicate that students viewed the

problem situation from increasing levels of realistic perspective. At the beginning of

the modelling process, there was a disconnect between students’ mathematical

solution and the original situation and operated with numbers with a minimal

understanding of the problem situation. After multiple interventions from the

researcher, students were eventually able to draw on their lived-experiences and to


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view the problem situation from an increasingly real world perspective and generated

a realistic mathematical solution.

MASOMEH NEJAD

UNDERGRADUATE STUDENTS’ PERCEPTION OF TRANSFORMATION OF

SINUSOIDAL FUNCTIONS

Trigonometry is one of the fundamental topics taught in high school and university

curricula, but it is considered as one of the most challenging subjects for teaching and

learning. In the current study, two different theoretical frameworks have been used to

examine undergraduate students’ perception of transformation of sinusoidal functions.

The results show that students did not grasp fully the concept of transformation of

sinusoidal functions.

OI-LAM NG

COHERENT AND DIVERGING DISCOURSE IN MATHEMATICAL

ACTIVITIES WITH DYNAMIC GEOMETRY

This paper discusses the detection of coherent and diverging discourse in

mathematical exploratory activities using dynamic geometry environments (DGEs).

The data is drawn from a larger study examining patterns of communication within

pair-work mathematical activities using touchscreen DGEs. A

thinking-as-communicating approach is used to analyse communication involving

pairs of high school calculus students’ exploration of the area-accumulating function

using touchscreen DGEs. Results show that the students integrated speech, gestures

and touchscreen-dragging synchronically and diachronically to engage in coherent

and diverging discourses. In particular, it is shown that conflicting ideas can be

communicated non-linguistically. This paper raises questions about new forms of

communication mobilised by touchscreen DGEs.

TANYA NOBLE

IDENTITY AND MATHEMATIS: A PRELIMINARY EXPLORATION

This preliminary exploration considers the Affinity-identity students have with

mathematics. The relevance of this became apparent as I noticed my own

Affinity-Identity reaffirm my growth into the role of a mathematics education

researcher. I explore possible ways to identify affinity and develop a picture of how

identity of the learner and the subject to be learned can be identified. Development of

identity is a communicative practice situated in cultural experiences and therefore an

analysis of both praxis and individual communication could offer deep insight into


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influences on student identity as they seek affinity with the identity projected onto

mathematics.

ZAKIEH PARHIZGAR, HASSAN ALAMOLHODAEI

STUDENTS’ FLOW EXPERIENCES IN THREE TYPES OF MATHEMATICAL

PROBLEMS AND TWO DIFFERENT EDUCATIONAL ENVIRONMENTS

In this study, flow experiences of 244 students in three types of mathematical problems

(intra-mathematical problems, word problems and modelling problems) were

examined. Pretest and posttest groups were used in this study. The treatment was that

the students, in two groups with different training formats (teacher-centered and

student-centered), attended 6 sessions of modelling problem solving. The results of this

study show that students’ flow experiences don’t have a significant difference in these

three types of mathematical problems. Teaching modelling problems increased

students’ flow experiences in reality-related mathematical problems, but it didn’t have

any effect on students’ flow experience in intra-mathematical problems. Also, the

student-centered group had positive effects on flow experiences in word problems.

SHEREE RODNEY

“THE OTHER TEN”: AUDEN’S SENSE OF NUMBER

This paper presents a case study of a student (age 5 years 6 months) name Auden, who

interacted with a touchscreen App called TouchCounts. This App was designed to

support children’s activities around counting. I use Sfard’s commognitive framework

to show how Auden thinks and learns about numbers. I show how Auden’s exploration

of numbers helps me understand the challenges children face when moving from

identifying subsequent number as they appear in the natural counting sequence to

identifying numbers that appears before and after each other.

ANNETTE ROULEAU

TEACHER TENSIONS: THE CASE OF NAOMI

Tensions are endemic to the teaching profession. Viewed as dichotomous forces,

tensions shape the experiences of mathematics teachers, affecting both their practice

and professional growth. In this article, I use Berry's (2007) framework to identify and

examine some of the tensions experienced by Naomi in her practice of teaching

mathematics. While previous research presents the image of teachers as dilemma

managers who accept and cope with continuing tensions, my research suggests that a

desire to resolve these tensions may impact teaching practice and professional growth

needs.


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MINA SEDAGHATJOU

A BLIND UNDERGRADUATE STUDENT’S JOURNEY TOWARD AN

UNDERSTANDING OF PRE-CALCULUS CONCEPTS

In this paper we argue that although mathematical communication and learning are

inherent multimodal and embodied subjects, sight disabled students are also able to

conceptualize visuospatial information. Adapting Vygotsky’s mediation theory, we

show that the lack of access to visual fields in an advanced mathematics course does

not obstruct a blind student in visualization, but rather modifies it. We argue proper

precise tactile materials might empower blind students to better visualize

mathematical functions.

JEFFREY TRUMAN

MATHEMATICS LEARNING AMONG UNDERGRADUATES ON THE AUTISM

SPECTRUM

This study examines the mathematical learning of an undergraduate student on the

autism spectrum, as well as some views of autism in the Vygotskian framework. I aim

to expand on previous research, which often focuses on younger students in the K-12

school system. I have conducted a series of interviews with one student, recording

hour-long sessions each week. The interviews involved a combination of asking for the

interviewee's views on learning mathematics, self-reports of experiences (both directly

related to courses and not), and some particular mathematical tasks. I present some

preliminary findings from these interviews and ideas for further research.

MILICA VIDENOVIC

OVERVIEW OF RESEARCH ON STUDENTS’ VIEWS OF ORAL ASSESSMENT

IN MATHEMATICS

With the increased emphasis on closed book written examinations, there is a critical

need for implementing the oral assessments in mathematics courses. Based on the

overview of research on students’ views of oral assessment in their mathematics

courses, most of the students had positive views towards its use. In this paper, the

research on oral assessment in mathematics was summarized with a specific focus on

students’ views.

MEDS-C 2015 Allan

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TEACHERS TEACH, STUDENTS LEARN?

Darien Allan

Simon Fraser University

This paper approaches the issue of student learning from the student perspective.

Leontiev’s activity theory is used to analyse observed student behaviours and student

goals to deduce student motive. In this study three student amalgams are analysed in

the particular activity setting of taking notes. Although the observed actions are

similar, the results show that the primary motives are different, and that this ultimately

had consequences for their learning. Thus, what teachers notice in the classroom may

not provide enough information to determine whether or not learning will occur.

INTRODUCTION

Substantial research has and continues to look at how to improve teaching, and how to

effect better student learning. What is often overlooked is a vital element: student

motive. Though not sufficient in itself, motivation is a critical factor in the

determination of the nature of the learning that occurs. The intent of this paper is to

approach the issue of student learning from the student perspective. Using Leontiev’s

activity theory observed student behaviours and student goals are analysed to deduce

student motive. The motives students hold affect whether or not learning occurs, and

the nature of that learning that does occur.

The aim of this particular study, and of the larger research project, is to investigate and

answer two questions: What are the behaviours that students exhibit in different

activity settings in the mathematics classroom; and What are the motives that drive

their behaviours? In this particular paper the observed behaviours take place in the

particular activity setting of ‘taking notes’. To address the first question, regarding

student behaviour, it is necessary to introduce the notion of ‘studenting’.

Studenting

The term ‘studenting’ was coined by Gary Fenstermacher in 1986. Initially, he

describes this concept in terms of a cohort of student behaviours including “getting

along with one’s teachers, coping with one’s peers, dealing with one’s parents about

being a student, and handling the non-academic aspects of school life” (p. 39). In

essence, Fenstermacher describes studenting as what students do to help themselves

learn. A later definition encompasses other behaviours such as “‘psyching out’

teachers, figuring out how to get certain grades, ‘beating the system’, dealing with

boredom so that it is not obvious to teachers, negotiating the best deals on reading and

writing assignments” (Fenstermacher, 1994, p. 1) and other similar practices.

Analyses have shown that the process of schooling produces a number of unintended

consequences, some desirable, but also many that are patently objectionable

(Engeström, 1991) and counterproductive to the goal of student learning. This appears

MEDS-C 2015 Allan

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to be especially true within the mathematics classroom. Preliminary studies have

shown that across the board students are finding ways to subvert the expectations of the

teacher in ways that the teacher is not aware of (Liljedahl & Allan, 2013a; 2013b).

Behaviours within the mathematics classroom

Although some student behaviours are ubiquitous across subjects, certain actions (or

inactions) may be more or less prevalent, or even unique to the secondary mathematics

classroom. Aaron and Herbst (2012) use ‘instructional identities’ to define and discuss

the particular attitudes students hold towards specific subjects and the behaviours they

enact within the confines of different instructional settings.

Within particular instructional situations, students may differentiate by taking on different

identities with different stances toward the work required by or toward the stakes of the

situation. … Because different students understand the figured world in different ways

these students feel that different actions are appropriate when faced with a mathematical

task. (p. 6)

Students come to the mathematics classroom with particular expectations about

mathematics and school. These expectations are reflected in attitudes towards and

behaviours within the mathematics classroom. I see instructional situations as a subset

of activity settings – one in which instruction is taking place, and one in which learning

is a presumed outcome.

If we entertain the idea that students behave differently in different instructional

settings (or activity settings) within the mathematics classroom (and in comparison to

other subjects), we are compelled to investigate the nature of these particular

behaviours, and in what specific settings they occur (or don’t occur). Furthermore, how

can we analyse what is driving students’ actions?

THEORETICAL FRAMEWORK

According to Leontiev, all activity is driven by some motive, whether the subject is

aware of that motive or not.

Activity Theory was chosen as a framework for analysis for its ability to describe what

is rather than what is ideal. Rather than categorizing students by their behaviours (e.g.,

on-task or off-task), presuming students behave rationally or ideally (e.g., game

theory), or assuming the students’ primary goal is learning (e.g., didactic contract

(Brousseau, 1997)), activity theory allows for a description of what is observed and

said without overlaying pre-existing assumptions or judgments. These observations,

taken together, can then be used to develop a hypothesis for what is driving student

action: their motive, which may be something other than a desire to learn. Individual

behavior and motive within the classroom collective is best viewed through a

theoretical lens primarily comprised of Leontiev’s Activity Theory (1978).

For Leontiev, “[a]ctivity does not exist without a motive; ‘non-motivated’ activity is

not activity without a motive but activity with a subjectively and objectively hidden

motive” (1978, p. 99). The object of an activity is its motive, and is something that can

MEDS-C 2015 Allan

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meet a need of the subject. Motives arise from needs, which are the ultimate cause of

human activity. Figure 1, below, illustrates the relationship between the elements in

Leontiev’s development of activity theory.

Figure 1: Pyramid Representation of Leontiev's Three-Level Model of Activity (1974)

Motives drive activity, and activities are directed at goals. People have many goals,

which shift in importance and in content on the basis of both contextual and

intrapersonal factors. At the very top of this hierarchy is the motive for activity. At any

time an individual has a hierarchy of these motives, the order of which is determined

through and as a result of one’s activity.

Actions are the many steps that comprise an activity, although not all are immediately

related to the motive (Kaptelinin & Nardi, 2012). Actions are directed towards specific

targets, called goals. Goals are conscious, in contrast to motives, of which a subject is

not usually aware.

The fact that motive is often hidden from the subject suggests difficulty in determining

the ultimate motive. This obstacle can be overcome by utilizing an “actions first”

strategy (Kaptelinin & Nardi, 2012). The strategy begins at the level of goals, which

people are generally aware of and can express, and the analysis is subsequently

expanded up to higher goals and ultimately the motive. Given the primary interest of

this study is the students’ actions/goals and activity/motive only the top two levels of

Leontiev’s pyramid will be considered in this paper.

Cataloguing the observable aspects of studenting (actions) and analysing these

together with students’ goals through the lens of activity theory can offer new insights

into student motives and provide researchers and educators with evidence to better

understand student behaviour.

Whether the behaviours discussed in the later section are unique to mathematics is

beyond the scope of this paper. Regardless, the issue of exclusivity with respect to

mathematics is not what is significant – what is important is that these behaviours are

occurring within the mathematics class, arguably much more frequently and

pervasively than the teacher perceives or suspects. These actions together with the

associated goals, point to particular motives that have significant impact on the nature

of the learning that occurs, or whether learning takes place at all.

MEDS-C 2015 Allan

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Although considerable research can be found looking at students’ motives and

behaviours, what is lacking is a catalogue of student behaviour across activity settings

in the mathematics classroom. This is the first objective of the larger research study,

the second is to investigate what drives this behaviour by analysing it using Leontiev’s

activity theory. This particular study looks at student behaviour in the activity setting

of ‘taking notes’.

METHODOLOGY

The nature of the research questions requires a particular approach – an ethnographic

study, though the study itself is not ethnography. In accordance with this approach, I

spent significant time immersed in the classes under study observing and interacting

with students, taking fieldnotes, and asking questions. Analysis occurred throughout

the process of data collection whereby what was observed and recorded in one lesson

provoked questions and shifted focus for the next observation/interviews. Both an etic

and emic perspective were utilized. In line with Mason (2011) I endeavour to provide a

strict ‘accounting of’ student behaviour rather than ‘accounting for’ observed actions.

Participants

The data for analysis are taken from a larger study conducted in three secondary school

mathematics classes in British Columbia.

Data collection and analysis

Data were collected during the 2013-2014 school year. Throughout the fall semester

the class was observed for twelve periods, each period ranging from 60 to 75 minutes.

Classroom lessons and informal interviews were audio recorded and transcribed for

later analysis and comparison with field notes taken during the class. The data

discussed here has been subjected to an analysis using Leontiev’s activity theory in

order to determine the likely primary motive underlying the students’ behaviour.

I have constructed profiles of three fictional students in order to illustrate potential uses

of Leontiev’s Activity Theory applied to studenting in the mathematics classroom.

These accounts are not fictional in the sense that all behaviours and goals were present

in the data, but do not belong to one unique participant in the study. I use this

amalgamation of cases because it is not possible due to space considerations to

describe all students who displayed particular types of behaviours. The amalgamation

of cases has been used previously by Piaget (1923/2001) and in mathematics education

by Leron and Hazzan (1997), Zazkis and Koichu (2014), and Liljedahl, Andrà, Di

Martino, and Rouleau (2015).

The profiles are created from a composition of student behaviours across the three

classrooms to demonstrate the similarities of the behaviours and goals across teaching

methods and schools.

MEDS-C 2015 Allan

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Although the students who comprise the profiles are from both genders, I have used

feminine names for the fictional students to ward against any gender stereotyping or

comparison.

RESULTS AND ANALYSIS

Profiles: Actions and goals

The three composite students described in this section are named Jenna, Kyla, and

Laura. All three students demonstrated the behaviour of taking notes in the activity

setting of a lesson. In what follows I first provide excerpts from informal interviews

with the students, then describe each student’s goals based on these conversations.

After this I analyse their motive(s) in the subsequent section. I chose the context of

taking notes, and consider only students who take notes because the observable

behaviour of taking notes is often perceived as a proxy for learning.

Jenna

Jenna copies down what the teacher writes on the board. When asked why she takes

notes, Jenna replied that it keeps her engaged. If she didn’t she might “lose focus and

doze off in class” and she has nothing else to do because “I’m not allowed to use my

electronic devices”. Jenna admitted “sometimes when I’m writing stuff down I feel like

I’m just writing it down, not actually learning it”.

Jenna appears to take notes because that is what is expected and to do anything else

might get her into trouble with the teacher.

Kyla

She writes notes so that she can “know how to do it for the booklet” if she forgets she

can refer to her notes “over and over until you get it”. Kyla says for review she “just

sits there and reads over the notes” but will also “refer to it if I have a question that’s

similar”. But if she has “homework that doesn’t really have to do with the lesson then

the notes won’t really help”. She feels that she “won’t remember anything if she

doesn’t write it down” and she believes that “the more you write the more you

remember because you’ve physically actually written the words”. Kyla looks at her

notes before the test to help her ‘remember’.

Kyla’s rationale for taking notes is characterized by an emphasis on memorizing or

remembering.

Laura

Laura stated “I feel like I understand it more if I write it down”. She writes

comprehensive notes including, but not limited to, what the teacher writes and says.

She later rewrites them outside of class: “When I take my notes I take them in complete,

exactly, everything the teacher says and then some, and then I add my own things to

them. And that’s how I comprehend most of the stuff … just if I can think of a way to

relate this to something else, or if there’s just an easier way of me thinking about it, I’ll

put that up by the side.”

MEDS-C 2015 Allan

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Laura writes down the teacher’s explanation (not just what is written on the board) and

later synthesizes her notes and adds to them, bringing in other ideas that she feels relate

to the topic and will help her to better understand it.

Motive

Looking at each student’s actions and goals, it is possible to expand upwards and

suggest her motive.

Considering Jenna’s stated reasons for taking notes, it appears that her goal during an

activity where she might be expected to take notes is to avoid getting into trouble.

These types of goals are associated with avoiding negative attention. Jenna’s primary

motive in this activity setting is not to learn, but to stay out of trouble and to ‘get

through’ the course.

Kyla appears to want to learn. However, it is not the same type of learning that one can

see in the description of Laura. Kyla is concerned with remembering, memorizing, and

not forgetting. She uses notes to help her complete similar homework questions and to

refresh her memory before a test. While one might suggest that her primary motive is

learning, the description of Kyla brings to mind Skemp’s ‘instrumental understanding’.

Kyla’s actions are most likely to result in her having procedural knowledge - knowing

only what to do or having “rules without reasons” (Skemp, 1987, p. 153). Her primary

motive is most likely to want to be seen as a good student and to get a good grade.

Laura, like Kyla, wants to learn. However, in contrast to Kyla, Laura is focussed on

understanding, in the sense of Skemp’s ‘relational understanding’. Rather than talking

about remembering or memorizing she tries to build connections between ideas.

Laura’s primary motive is learning.

For me, learning and understanding is relational understanding – knowing not only

what to do, but why to do it. In this vein, a student can only have a primary motive of

learning when it arises from goals linked to relational understanding. Only Laura has a

primary motive of learning and thus only Laura is likely to have an outcome of

learning.

DISCUSSION AND CONCLUSION

The goals and motives students hold have consequences for their learning. On the

surface, with respect to their observed behaviours within the classroom, each of the

three students performed similar actions. So to the teacher, each of these students is

doing what is expected (whether it was explicitly required or not). Underpinning their

observable actions are students’ reasons for taking notes, their goals, and this is where

the students diverge. The reasons for taking notes that the students offered in the

informal interviews point to the students’ goals, which can in turn be used to expand

upwards to ultimately determine motive. Jenna’s motive was to stay out of trouble,

Kyla’s was to get a good grade, and Laura’s was to learn. The particular motive a

student holds is important because if a student does not have a primary motive of

learning, the outcome will be something other than learning. Although students’

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achievement was not explicitly measured during the study, it can be reported that Jenna

performed poorly on assessments, Kyla had satisfactory results, and only Laura

demonstrated real understanding of the concepts.

It is important to remember that all of these students’ observable actions appeared the

same. From the teachers’ perspective, things like taking notes, doing homework, and

coming for help are indicators or stand-ins for learning and understanding – even if

they are not explicitly required. Although teachers may see students’ actions as

indications of learning, they are often just proxies for learning. Thus even when

students are conforming to (possible) teacher expectations, as in the three student

amalgams, there appears to be a gap between actual student activity and (relational)

understanding, the ideal outcome associated with a primary motive of learning.

Frameworks like the didactic contract (Brousseau, 1997) may apply when the desire to

learn drives student activity but when learning is not the primary motive, other

approaches must be explored, such as activity theory.

Other preliminary results show that there are a variety of goals that manifest in similar

actions, suggesting that considered alone, a student’s actions are insufficient to deduce

his or her motive. These results also show that even when students do not conform to

the teacher’s expected behaviours, from the student’s perspective there is a certain

rationality to their actions. Although the analysis of the results is only in the initial

stages, it is evident that in many cases non-conforming studenting does have a valid

justification. It is anticipated that extended analysis will provide further evidence to

support this claim.

What these results suggest is that a deeper understanding of the perspective of the

student within the classroom unit could serve to provide teachers with cause for

reflective thought on their policies and practices that influence student learning.

References

Brousseau, G. (1997). Theory of didactical situations in mathematics. N. Balacheff, M.

Cooper, R. Sutherland, & V. Warfield (Eds.). Dordrecht, Netherlands: Kluwer Academic

Publishers.

Engeström, Y. (1991). Non scolae sed vitae discimus: Toward overcoming the encapsulation

of school learning. Learning and Instruction, 1, 243-259.

Fenstermacher, G. (1986). Philosophy of research on teaching: Three aspects. In M.C.

Whittrock (Ed.), Handbook of Research on Teaching (3rd ed., pp. 37-49). New York, NY:

Macmillan.

Fenstermacher, G. (1994, revised 1997). On the distinction between being a student and

being a learner. Paper presented at the annual meeting of the American Educational

Research Association, New Orleans, LA.

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Kaptelinin, V., & Nardi, B. (2012). Activity Theory in HCI: Fundamentals and reflections

(Synthesis Lecture Series, John M. Carroll (Ed.), Penn State University). San Rafael, CA:

Morgan & Claypool.

Leron, U., & Hazzan, O. (1997). The world according to Johnny: A coping perspective in

Mathematics Education. Educational Studies in Mathematics, 32, 265-292.

Leontiev, A. (1974). The problem of activity in psychology. In J. V. Wertsch (Ed.), The

concept of activity in Soviet Psychology (pp. 37-71). New York: Sharpe.

Leontiev, A. (1978/1975). Activity, consciousness, and personality. Englewood Cliffs, NJ:

Prentice Hall.

Liljedahl, P., & Allan, D. (2013a). Studenting: The case of "now you try one". In A. M.

Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference for the International

Group for the Psychology of Mathematics Education (Vol. 3, pp. 257-264). Kiel,

Germany: PME.

Liljedahl, P., & Allan, D. (2013b). Studenting: The case of homework. In M. V. Martinez &

A. Castro Superfine (Eds.), Proceedings of the 35th Conference for the Psychology of

Mathematics Education – North American Chapter (pp. 489-492) Chicago, USA:

PMENA.

Liljedahl, P., Andrà, C., Di Martino, P., & Rouleau, A. (2015). Teacher tension: Important

considerations for understanding teachers’ actions, intentions and professional growth

needs. In Proceedings of the 39th Conference for the International Group for the

Psychology of Mathematics Education. Hobart, Tasmania: PME.

Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. R. Jacobs, & R. A.

Phillipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35-50).

New York: Routledge.

Piaget, J. (1923/2001). The language and thoughts of children. London, UK: Routledge.

Skemp, R. (1987). The psychology of learning mathematics: Expanded American edition.

Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Zazkis, R., & Koichu, B. (2014). A fictional dialogue on infinitude of primes: introducing

virtual duoethnography. Educational Studies in Mathematics, 88(2), 163-181.

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AN EXPLORATION INTO A SAUDI MATHEMAICS TEACHER’S

FIGURED WORLDS

Lyla Alsalim


In this paper, patterns-of-participation theory serves as a lens to interpret and

understand Saudi high school mathematics teachers’ practices. This framework

focuses mainly on understanding what practices and figured worlds are significant for

the teacher and how the teacher engages in those figured worlds. The data presented is

about Abeer, a high school mathematics teacher in Saudi Arabia. The data generated

suggests that there are five significant practices or figured worlds to Abeer’s sense of

her practice as a mathematics teacher. The paper discusses and explains these figured

worlds.

PURPOSES OF THE STUDY

In Saudi Arabia, the education system has undergone major changes in the past decade.

Government agencies involved in education have introduced new policies, standards,

programs, and curricula. One of the major reform initiatives directly addresses existing

mathematics curriculum. In 2011, the Ministry of Education introduced new

mathematics textbooks, which is usually the primary, and sometimes only, resource for

teachers. The new mathematics textbooks are based on the curricula published by

McGraw Hill Education Learning Company. This is the first time that the Ministry of

Education introduced textbooks that were developed outside the country. These

changes are accompanied by high expectations that teachers will incorporate the

changes seamlessly without consideration of their existing practices. This paper is part

of an ongoing study that intends to gain a better understanding of how high school

mathematics teachers in Saudi Arabia are coping with recent education reform,

including how their practices are evolving in response to the changes that are

happening in the education system.


In this paper, patterns-of-participation (PoP) (Skott, 2010, 2011, 2013) approach

serves as a lens to interpret and understand Saudi high school mathematics teachers’

practices during the current reform movement. The PoP framework identifies teachers’

practice as being how teachers narrate and position themselves in relation to multiple,

and sometimes conflicting, figured worlds (Skott, 2013). Figured worlds are imagined

communities that function dialectically and dialogically as if in worlds. They constitute

sites of possibility that offer individuals the tools to impact their own behaviour within

these worlds (Holland, Skinner, Lachicotte, & Cain, 1998; Skott, 2013).

Traditionally, most research in education that focuses on studying teachers’ practices

adopt an acquisitionist approach, especially those studying teachers’ beliefs and

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knowledge in relation to teachers’ practices (Skott, 2013). Recently, more researchers,

including Skott (2009, 2010, 2013), adopt participationism as a metaphor for human

functioning to understand teachers’ practices. “The origins of participationism can,

indeed, be traced to acquisitionists’ unsuccessful attempts to deal with certain

long-standing dilemmas about human thinking” (Sfard, 2006, p.153). Skott presents

PoP as a coherent, participatory framework that is capable of dealing with matters

usually faced in the distinct fields of teachers’ knowledge, beliefs, and identity.

Therefore, PoP is a theoretical framework that aims to understand the relationships

between teachers’ practice and social factors. Skott (2010, 2011) initially developed

the patterns-of-participation framework in relation to teachers’ beliefs. However, in

order to develop a more coherent approach to understand teachers’ practices, Skott

(2013) extended the framework to include knowledge and identity.

The social approach of research in mathematics education has progressively promoted

the notion that practice is not only a personal individual matter; it is in fact situated in

the sociocultural context. Although the relationships between individual and social

factors of human functioning have generated much debate in mathematics education, it

is mainly in relation to student learning (Skott, 2013). Therefore, PoP is a theoretical

framework that aims to understand the relationships between teachers’ practice and

social factors. To a considerable degree, PoP adopts participationism as a metaphor for

human functioning more than mainstream belief research. Therefore, PoP draws on the

work of participationism researchers, specifically Vygotsky, Lave & Wenger, and

Sfard.

“The intention of PoP is to take this one step further by limiting the emphasis on

acquisition and include a perspective on the dynamics between the current practice and

the individual teacher’s engagement in other past and present ones” (Skott, 2013, p.

557). This framework focuses mainly in understanding what practices and figured

worlds are significant for the teacher and how the teacher engages in those figured

worlds. A teacher’s engagement with these figured worlds inform and adjust the

interpretations s/he makes to her/himself and the way s/he engages in on-going

interaction in the classroom. These figured worlds work in a very complex system

where they could support and sometimes, restrict one another as the teacher contributes

to classroom practice.

METHODOLOGY

This paper is part of an ongoing study that intends to develop more coherent

understandings of Saudi high school mathematics teachers’ practices during the

current reform movement. For data analysis, I used a qualitative analysis approach

based on grounded theory method as used by Skott (2013).

The data presented in this paper comes from Abeer, a high school mathematics teacher.

Abeer has six years of experience working at the same school. She graduated from

university with a Bachelor of Education degree with a specialization in mathematics.

The education courses Abeer had in university focused on general issues related to

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teaching, such as lesson planning and classroom management. She does not have any

experience taking educational course related to teaching mathematics specifically.

Abeer was not among the teachers who used the new textbooks in the first year of

implementation since she was teaching grade 11 at that time. She has been teaching

from the new textbooks for three years. When I met Abeer she was teaching 21 lessons

per week to students in grade 11.

I conducted two semi-structured interviews with Abeer. The first one before I observed

her teaching two lessons and the second interview was conducted after the classroom

observations. In the first, I asked her to reflect on her experiences with mathematics

teaching and learning in school, at university, and during her practicum year. I also

asked her questions related to different aspects of the new reform movement in

education system in Saudi Arabia. I also asked Abeer to reflect on her experience

teaching mathematics using the old and the new textbooks. The second interview

focused on her experiences with teaching mathematics at her school and on her

relationships with the school, her colleagues, and the students. I asked her to reflect on

the lesson planning process she had in order to prepare the lessons I observed. During

the second interview, I also used a stimulated recall technique by playing audio

recordings of parts from the lessons to facilitate her conversation about her own

teaching practice in the classroom.

During my visits to Abeer’s school, I was also able to collect some data from informal

observations of staff-room communication between Abeer and her colleagues. I also

have a copy of Abeer’s lesson planning notebook and some of her worksheets and tests

samples.

DISCUSSION

The aim of this paper is to develop a deeper understanding of the participant teacher’s

significant practice and figured worlds and how she engages with these figured worlds.

As a teacher positions herself in relation to her profession as a mathematics teacher,

she draws on several, often incompatible, figured worlds. Her engagement with these

figured worlds does not only appear in her verbal communication, but also by the

choices she makes in her all other actions related to her profession, such as her

immediate reaction to certain student behaviors or the way she expresses her view

when engaging in a conversation with her colleagues.

Teachers’ engagement with figured worlds inform and adjust the perceptions they

make and the way they engage in on-going interaction in the classroom. These figured

worlds work in a very complex system where they could support, and sometimes

restrict, one another as every teacher contributes to classroom practice.

Abeer’s classroom

Abeer has 30-33 students in every class. Her students sit in groups that do not remain

static; she changes who is in each group every week. New groups are formed every

Sunday (the first day of the school day in Saudi Arabia) by assigning students

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randomly. The groups are arranged in square formation with four to five students in

each group. In her classroom, students work individually as well as in groups. The

textbook is always a part of her lesson. She usually starts her lesson with an activity

that students do within their groups. She often chooses the activity from the textbook,

but sometimes she comes with a different activity than what is in the textbook. Abeer

rarely assigns her students homework, but when she does, her students know that they

are not required to work on it at home. Every Thursday, which is the last school day of

the week, the lesson starts with a quick quiz. The quiz is often related to what they have

been doing in class during the week.

RESULTS

The data generated from Abeer’s interviews suggest that she has five significant

practices or figured worlds in her practice as a mathematics teacher after six years of

teaching. These figured worlds are mathematics, textbooks, responsibility for students’

achievement, reform or change, and her relation with other in school. It is important to

clarify that I am not claiming that these are the only figured worlds that contributes to

Abeer’s sense of her practice as mathematics teacher.

Mathematics

Abeer’s sense of her practice has developed as a result of her own experiences as a

student and a teacher of mathematics. From her experience as a student, she was

always told that learning mathematics is about finding the right answer to a problem.

Her own notion of mathematics learning as a teacher is quite different. “One of the

most important aspects of mathematics learning is encouraging students to focus more

on the thought processes being used rather than focusing completely on finding the

correct answer”. When Abeer decided to become a mathematics teacher, she knew

there were two ways to do so, the “easy way” and the “hard way”. “It would be an easy

choice to teach mathematics as a subject of right and wrong answers and that's all there

is to it”. Abeer decided to take the hard challenging way.

Doing mathematics in Abeer’s classroom is not about finding the right answer, it is

about applying thinking skills and being able to explain how to find an answer. She

endorses students' ability to think reflectively about their answers. Abeer’s ultimate

goal for her students learning experience is to get her students to get to a point where

they are sure of their answer and they can explain how they got it.

In her classroom, Abeer often invites her students to think of multiple approaches to

solving mathematical problems because, as she indicates, it is very important to

student learning of the mathematical concepts. Having students look for multiples

ways of finding solutions and using multiple strategies provides Abeer with more

opportunities to encourage her students to talk about their work and explain how they

reached a solution.

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The textbook

Abeer has a strong appreciation for the textbooks she is currently using in her teaching.

The implementation of the new textbooks, according to Abeer, was a necessary step.

For her, the old textbooks were insufficient in providing the knowledge and incentives

for teachers to reflect on their practice. Abeer uses the new textbooks as a tool to

reflect on her practice. She indicates that when she plans her lessons, and before she

chooses a particular part from the textbooks to use in her lesson, she asks herself,

"What do I actually want to get away from this? What do I need my students to take

away from this? How can I add to enrich students learning experience? By answering

these questions, I can imagine different scenarios.” It seems that Abeer engages deeply

with the textbook during her planning for the lesson phase. Also, in her classroom

Abeer often invites her students to engage with the textbook.

In her classroom, the textbook has an active presence. Abeer often invites her students

to engage with the textbook. At the beginning of the lesson, she asks the students to

read the parts of the lesson, which are previously covered skills and concepts, learning

outcome of the lesson, and the major mathematical vocabulary used in the lesson. She

gives them few minutes to discuss what they read in their groups. Then, she asks them

to read the purpose of the lesson section, which presents information usually related to

real life situations and sometimes requires the students to answer questions that follow

the information. She usually discusses this part with the students and this discussion

leads to the introduction of the main concept of the lesson.

During the lesson, Abeer also refers her students to the textbook many times. The

students read most of the instruction part of the lesson. Sometimes, Abeer reads certain

parts of the textbook and the students follow along in their textbook; other times, she

asks one student to read aloud from the textbook, and in some cases the students are

asked to read some parts and discuss in their groups.

The reform or change

Abeer is very enthusiastic about the current reform movement in Saudi Arabia. Her

enthusiasm shows from her involvement in any professional development activities

offered to her. She indicates that most teachers don’t like to participate in any activity

related to professional development. For Abeer, these activities provide her the

opportunity to find new ideas, meet different teachers, and talk about mathematics

teaching. Abeer is excited and strongly supports the new curriculum, which is

considered one of the major steps in the reform movement to the education system in

Saudi Arabia. For Abeer, the new mathematics curriculum created a new language to

talk about mathematics teaching and learning.

Reform for Abeer means change. When she talks about reform she usually explains

how the current reform movement has changed some aspects of her practice. Abeer

described the first two years of her teaching career as being very traditional. “I was the

person who had the mathematics and my job was simply to present what I know in the

class. I used to start my lesson by presenting the concept and writing on the board the

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definition and the formula, and then I would work few examples and then give them a

worksheet and tell them to do just as I did then I would correct them really quick”.

Abeer re-defined her role as teacher in the classroom as a result of her inspiration of the

reform ideas.

Abeer’s relationship with other staff members in her school

Abeer appreciates the experience of working in her school. She is dedicated to the

development of her school. She realizes the importance of a school community to

create a positive learning environment for everyone. She indicates that schools are not

only for sitting and learning for students; “everyone who works at this school learns

from this experience”. Abeer values communication with other teachers in the school

and considers it an important source for her personal learning experience as teacher.

Abeer also draws on the principal’s support for a learning community among teachers.

According to Abeer, although the principal of the school was not a mathematics

teacher, she appreciates mathematics as a subject and realizes its importance.

A school inspector is a part of the education system in Saudi Arabia. . For Abeer, the

inspector’s visit is usually a challenging part of her practice. Her general view of what

is valuable in her teaching does not seem to match the inspector’s view. According to

Abeer, “the inspector cares about how much I give and whether I am following the

plan we received from the Ministry of Education. I don’t follow their plan; I usually

make my own plan which I always adjust during the year.”

Responsibility of students’ achievement

In the interviews, Abeer appears strongly committed to her students’ achievement.

Sharing students’ main interest and understanding their biggest concerns is significant

to her philosophies about her role as a high school mathematics teacher. For Abeer,

teachers are meant to meet the needs and address the concerns of their students. Abeer

constantly engages with her students’ emotional state of being under pressure to

achieve in high school and then move on to college. Abeer relies on the collaborative

working environment she has at her school to reflect on students’ achievements.

During her group meetings, Abeer talks with other teachers about improving students’

achievement and helping them achieve not just at school but on other standardized tests

they take during high school. She and the other teachers organize workshops for

students to help them prepare for these tests.

CONCLUSION

Abeer has a strong dedication to her job and a willingness to deal with the challenges

and the responsibilities involved. She has always wanted to become an educator and

was greatly influenced by many of her experiences learning mathematics at school.

Her main motive to become mathematics teacher was that she wanted to be able to

share with her students the amazing rewarding experience learning mathematics can

have. According to Abeer, interacting with the students is the best part of being a

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teacher. The most exciting moments are when you can actually see students start to

understand mathematics concepts.

As Abeer positions herself in relation to her profession as a mathematics teacher, she

draws on several, often incompatible, figured worlds. Her engagement with these

figured worlds does not only appear in her verbal communication, but also by the

choices she makes in all other actions related to her profession such as her immediate

reaction to certain students’ behaviour, or the way she expresses her view when

engaging in a conversation with her colleagues. It is also important to clarify that I am

not claiming that these are the only figured worlds that contributes to Abeer’s sense of

her practice as mathematics teacher. It is very challenging to get access to all the

practices and figured worlds that are possibly significant for Abeer’s classroom

interaction. For instance, challenges could occur if the figured worlds are related to the

teacher’s experience in schools and university (Skott, 2013). I plan to collect more data

to gain a deeper understanding of Abeer’s teaching practices.

References

Holland, D., Skinner, D., Lachicotte, W., Jr., & Cain, C. (1998). Identity and agency in

cultural worlds. Cambridge, MA: Harvard University Press.

Sfard, A. (2006). Participationist discourse on mathematics learning. In J. Maasz & W.

Scholeglmann (Eds.), New Mathematics Education Research and Practice (pp. 153-170).

Rotterdam: Sense.

Skott, J. (2010). Shifting the direction of belief research: From beliefs to patterns of

participation. In M.F. Pinto & T.F. Kawasaki (Eds.), Proceedings of the 34th Conference

of the International Group for the Psychology of Mathematics Education (Vol. 4, pp.

193-200). Belo Horizonte, Brazil: PME.

Skott, J. (2011). Beliefs vs. patterns of participation – towards coherence in understanding the

role the teacher. In B. Roesken & M. Casper (Eds.), Proceedings of the 17th Conference of

Mathematical Views, 211-220. Bochum, Germany: MAVI.

Skott, J. (2013). Understanding the role of the teacher in emerging classroom practices:

searching for patterns of participation. ZDM Mathematics Education, 45(4), 547–559.

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A PHENOMENOLOGY PERSPECTIVE TO STUDY

PROFESSIONAL DEVELOPMENT

Melania Alvarez


The purpose of this phenomenological study is to provide a phenomenological

framework to study professional development.

INTRODUCTION

Extensive research has been conducted about students and pre-service teachers

learning in a variety of classroom settings where their interactions with instructors or

facilitators are examined, but this is not so much the case with in-service teachers as

they are engaged in learning opportunities provided by professional development. As I

was doing research regarding successful professional development, I found no

descriptions about the lived experiences between professional developers and teachers.

What I usually found were survey results from teachers’ program evaluations, which

provided information about teachers’ individual perspectives regarding the

effectiveness of professional development sessions, or statistical analysis of school

performances after the application of a particular professional development

opportunity. Not much if anything has been done to show what happens during those

learning opportunities and how teachers’ and professional developers’ predispositions

and the setting allow for a fruitful learning opportunity and when they will not (Garet,

Porter, Desimone, Birman, & Yoon, 2001; Grammatikopoulos, Gregoriadis, &

Zachopoulou, 2013; Linder & Simpson, 2014). This study presents a framework that

attempts to provide an initial tool for a description and analysis of this experience.

FRAMEWORK

The use of the phenomenological perspective seems to be quite suitable for the task at

hand, given that the purpose of this research is to analyze the “lived experience” of

professional development sessions and describe teachers’ reactions to a variety of

approaches and activities put forward by the professional developer. This study

explores and describes “lifeworld” learning experiences, and look for the meaning of a

phenomenon by uncovering, as much as possible, the many layers that socially and

culturally influence a person’s experience in their lifeworld, where lifeworld is defined

by Van Manen (1997) as “the world of immediate experience”, the world as “already

there” (p. 182). According to Woodruff (2013) phenomenologist practices consist of

three different methods:

Rich descriptions of lived experience

Use of relevant features in the context to interpret the experience

Analysis of the form of a type of experience

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One can combine all three to analyze the description of the lived experience, and to

interpret it by assessing and using the relevant features in the context and analyzing

structures which resonate with our own experience—that which one can be conscious

of. This study focuses on engagement, and how through engagement teachers show

their motivation, wants, needs and learning. In this study, engagement is defined as

different ways in which individuals and groups come into action, or become involved

with an activity, where these different ways of acting/engaging are influenced by

different levels of motivation, motives, and goals.

What is the difference between motivation and motive? According to the experts,

motivation is a theoretical construct that can be used to explain behaviour (Elliot &

Marti, 2001; Pardee, 1990), whereas motive is something that moves a person to do

something (Ryan & Deci, 2000). According to Ryan and Deci’s Motivation Theory

(2000), there are different levels and different types of motivation which are closely

related to attitudes, goals, and/or wants that prompt an individual to particular actions,

and depending on the source of attitudes, goals, or wants we would be able to

distinguish between intrinsic and extrinsic motivation. Motive and wants are concepts

that point to elements within and outside of an individual, and they seem closely

connected. Motive corresponds to a want or a preference that is sufficiently strong

enough that it moves us to action or deliberate inaction (Braybrooke1992). In this

study a want is something we want to be given or we want to obtain and motives will be

something that would drive us to actions, with this action cast back to engagement,

which is usually an indicator of learning.

According to Woolfolk’s (1998) motivational theory, people's motives and wants are

behind the level of engagement, which is geared towards specific intentions or

objectives. Because motivation and motives are deeply guided by teachers’ wants,

these key elements will hinder or encourage teachers’ engagement and learning during

the professional development session, and the facilitators must tend not only to their

own motivation, motives or wants but also those of the teachers, while still planning

and delivering the professional development sessions and keeping as much as possible

the original objectives of the session. Research has shown that the motives and wants

that teachers bring into a professional development environment affect teachers' levels

of engagement during the session (Liljedahl, 2014). Consequently, engagement is an

indicator of wants and motives and vice versa. So, teachers' engagement in an activity

can provide a good source of information regarding their learning and their practice.

As mentioned before the literature mentions level of engagement as a way to find out

the success of a particular opportunity for learning. However, in the literature the term

“level of engagement” is used but not really defined. Woolfolk’s (1998) motivational

theory mentions intensity of involvement in an activity, but does not provide a

definition for level of engagement. But how is intensity of involvement defined?

There are two elements that usually allow us to sense how intensely involve

participants are in an activity, these two elements are flow and mood and that usually

will provide professional developers with signals as to how to proceed.

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This study will consider engagement as mainly composed of two factors: flow and

mood; where I define flow as the fluidity of action. Flow indicates a measure

(intensity) of action from an individual or a group. I define mood as state of feeling or

being at a particular time and place. Flow and mood are helpful and relevant tools in

analyzing engagement in trying to recognize when learning is happening through

communication and cognition and when it is not and why. Levels of flow in

communication and the mood among participants during professional development

sessions will indicate different types or modes of engagement, some of which are

conducive to learning and some which are not. Through engagement one can witness

moments of reflection, moments of enlightenment, exploration, rejection, resistance,

confusion, fear, rejection, doubt, participants’ motives and wants, and ultimately their

learning. Changes in flow and mood, can bring about points of inflection, which in

some instances will indicate a commognitive conflict, which can lead to or produce

learning occurrences for an individual or the whole group or where an opportunity for

learning can be lost. The commognitive adjective crafted by Sfard (2007, 2008)

implies that thinking is not a self-contained mechanism separate from speaking or

more generally communication but as a form that “have developed from a patterned

collective activity” (p. 571), and she views thinking as “an individualized form of

interpersonal communication” and as such “a product of collective doing” (Sfard,

2008). She sees commognitive conflict as the source of mathematical learning, and

which she defines as “the phenomenon that occurs when seemingly conflicting

narratives come from different discourses – from discourses that differ in their use of

words, in the rules of substantiation, and so on. Such discourses are incommensurable

rather than incompatible, that is, they do not share criteria for deciding whether a given

narrative should be endorsed… cannot automatically count as mutually exclusive even

if they sound contradictory… whereas acquisitionist views conflict resolution as

making sense of the world, commognitivist regard it as making sense of other people’s

thinking (and thus talking) about this world. This means gradual acceptance,

‘customization’, and rationalization—figuring out the inner logic – of other people’s

discourses.” (Sfard, 2007, pps. 577-578).

This framework which is based on a modified version of Remillard’s (2012) analytical

perspective, which connects positioning and engagement in order to analyze the

various ways a professional developer tries to position teachers in an experience that is

engaging to them. Professional developers want to position teachers in a way where

interaction is possible and furthermore this is something that they want to do of their

own volition. Remillard classifies positioning and interactions by using the following

terms, which I redefine as follows:

mode of address: how a group of people are position to engage in an activity.

forms of address: artifacts/resources used in order to position people.

modes of engagement: the way people react/engage

forms of engagement: what artifacts/resources, ideas result from the activity.

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The framework provides a circular model (Figure 1). It starts when the professional

developer plans and applies particular modes and forms of address during the

professional development session. The particular forms and modes of address run

within a particular background of goals, motives and wants from all the participants

involved including the professional developer. Particular mode of engagement follows

with particular moods and levels of activity or flow, and forms of address are

re-sourced into particular forms of engagement. As the professional developer

observes and reflects upon the resulting modes and forms of engagement, she responds

accordingly either by continuing or modifying the current mode and form of address.

Modifications depend on how much the professional developer is able to perceive a

level of learning where hopefully the teachers' and professional developer's motives,

wants, and goals are addressed and realized.

METHODOLOGY

So, how can I describe and put together all of these elements without simply creating

long, boring narratives? A visual analytical tool would be helpful and I was directed by

my advisor to look at interactive flowcharts. Flowcharts were introduced by Sfard

(2001) and Sfard and Kieran (2001) in order to allow the reader to experience

meta-discursive intentions in conversations (ibid. p. 58), and in our case it also would

allow us to visualize the fluidity or flow of the conversation. In Sfard and Kieran’s

(2001) charts interactions among participants are indicated by re-active, pro-active

arrows, in our model we have added arrows that indicate implicit agreement, support or

reflection. Additionally, arrows come in different colors to indicate the kind of

preoccupations of motivations behind the interaction. Data regarding the actions of

teachers in a professional development setting were gathered in order to describe their

participation and engagement throughout the sessions. This was done by

audio-recording most of the professional development sessions, surveys, and through

notes made by the facilitator during and after each session about teachers’ engagement.

After every session notes were made by the professional developer reflecting on the

effectiveness of the session, which concerns were addressed, the level of engagement,

her feeling and thoughts as well as questions that she wished she had asked during the

session and questions that she hoped to ask at the next session with the group. This

reflective process served as a preliminary analysis of the data (Glesne, 1999). Most

observations of teachers’ behaviour were made during the professional development

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sessions. For the data analysis the following phenomenological protocol was used,

which is a simplified version of Hycner’s (1999) process used by Groenewald (2004)

together with some steps delineated by Van Manen (1997): First, we investigated the

experience as we lived-it, then we bracketed the assumptions being made, as we looked

at the data we delineate units of meaning, then we clustered them in themes and we

create a composite. The participants in this study were the teachers involved in

professional development, and the facilitator. The professional developer, is also a

participant because her reflections and responses to teachers’ engagement as well as

my motives and wants are part of the study.

PRELIMINARY RESULTS

The following chart depicts an interaction with a group of Kindergarten teachers.

Before the following interaction the professional developer spent a substantial amount

of time discussing with them the resources and tried to make them look at particular

lessons and have a discussion about them.

Mode of address the professional developer asked teachers a questions that would

help them reflect on their previous practice and compare it with what was done in the

program that we were looking it. In this particular case, they have been looking at

lessons and activities within them to teach students words and their mathematical

meaning and use; words like same, different, more, less, etc. This particular program

emphasizes the importance of teaching one term at the time to make sure that students

will fully understand before using words together to describe an object.

Flow and Mood of engagement: The flow of the conversation change from being

slow as they were reflecting on the professional developer questions and then the

conversation became more fluid, the teachers became actively engage in analyzing the

lessons they were looking at. We can see this change as the arrows change, we first

have active pro-active and re-active together arrows, which indicate that there is not

much interaction among the teachers, however those arrows change to supportive and

reflective arrows which indicate a more fluid engagement. Jane’s and Cosima’s mood

is first rejecting (Jane first reaction was to say no) but then with Cosima’s reflective

answer it changes first to reflective and their engagement became mainly exploratory.

Units of meaning: Asking about comparing their current practice with what is

expected in the current program had an effect with these teachers because they could

see how this program could address some issues they were not aware of and that

possibly was interfering with their students learning.

The first two point of inflection are an AHA moment that will further trigger moment

of reflection making sense individually of a new realization about their practice and

how to improve it. From there an exchange of ideas took place among the teachers and

then I saw pertinent to ask a second question so that they will focus on the math. From

here a deep discussion between Jane and Cosima started where they discussed

possibilities, changes and adaptations to their practice

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Table K.2.4 Facilitator Jane Cosima Notes

Facilitator: Up to now each lesson explores one concept. How busy are usually your lessons? Do you teach one or several concepts at the same time? Jane: No, uh [not] really. Cosima: (reflecting and commenting) I do. I do because to me is so easy

O

O

O

O

O

O

O

O

O

Point of inflexion.

Jane (reflecting): I was not always aware. I don’t think I was aware that I am doing more than one concept, but this is so specific that it makes me recall right back from maybe. Jane is reflecting on her previous practice versus what she sees in the new program. She is talking to us but not really (this is marked by the dashed arrow)

O

O

O

O

O

O

O

O

O

Point of Inflection

Cosima: Even when we are doing addition...I am not following concrete, pictorial, abstract… I never realized it is a

different thing for them, to see it rather than touch it. (She was going through the same process of reflection as Jane)

O

O

O

O

O

O

O

Then a conversation ensued between Jane and Cosima about doing too much during a lesson and getting the children confused and how to watch out for that and check each other's lessons for a while for busyness. They were looking at the manual, discussing the topic in unit one.

O

O

O

Then, while looking at patterns I made the following comments: Facilitator: In order to be able to see patterns, perhaps we should look and describe the design. A discussion ensued about patterns and how to discuss different attributes in class. I did very little prompting afterward. Jane and Cosima were discussing a variety of ideas and they discussed what to do to help students articulate what they actually see. They also compared previous practice with what they saw of the new program.

O

O

O

O

O

O

Point of inflexion

Excerpt 1

Finally, they asked me several questions about how to use the resources for the lessons we just reviewed, but this time it was not about classroom management. Rather, we actually looked at the mathematical ideas, and also there was a discussion between Jane and Cosima about how to get the parents involved.

O

O

O

O

O

O

O

O

O

Afterwards without much prompting they looked at the following two lessons and they mainly had a discussion among themselves. They were carefully reading the teacher’s guide and asking questions. But the general diagram of the following 26 minutes looked like this. Either Jane or Cosima would ask me a question about terminology and/or the program and I would

O O O O

O O O O

O O O O

Excerpt 2

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reply and then they would have a discussion on their own. The thick black arrow indicates previous practice, new program, mathematical ideas and terminology.

Blue arrow: solid means we are discussing the mathematical part, and the dashed arrow means we are discussing classroom management with resources.

Red arrow: previous practice

Solid black arrow: New program, previous practice, and terminology

Green arrow: terminology and definitions

Orange arrow: parent opinions.

Wants: Cosima and Jane wanted a program that made sense to the children and had a

logically sound development and to be able to have a full understanding of

mathematical connections and sequencing. I wanted them to look carefully at the

lessons and have a discussion about the pedagogy and the mathematics involved which

they did.

Themes: Connecting a new program with previous practice can be a good way of

learning about teacher’s previous practice and position them in a way where they

actively become engage with the new program. A reflective mode can lead to an

actively exploratory mode.

CONCLUSION

Analyzing professional development under the phenomenological paradigm is a task

that has to take into account many aspects that are part of this activity; this framework

is a first step into that direction. In analyzing how the professional developer positions

participating teachers into an activity and analyzing their reaction or “level of

engagement” by looking into the flow, mood, the possible wants, the units of meaning

and themes it is our hope that we can better answer the question that many of us ask as

we develop our professional developing sessions: what works and how?

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“WHAT WE NEED TO SHOW IS THAT T IS WELL-DEFINED”:

GESTURE AND DIAGRAM IN ABSTRACT ALGEBRA

Andrew Hare

Simon Fraser University, Canada

Proving that a mapping is well-defined is a key step in many results in group theory.

Such an argument is required in the Orbit-Stabilizer Theorem, as well as the theorem

that gives a necessary and sufficient condition for the set of cosets of a subgroup to

form a group (namely, that the subgroup is a normal subgroup). It can be challenging

for students to even realize that such an argument must be provided, as it is the first

time they have been exposed to functions defined on equivalence classes. This paper

studies the gestures and diagrams that one professor uses to explain the concept of

well-definedness.

INTRUDUCTION

The first abstract algebra course that an undergraduate math major takes can be a

turning point for the student (Weber & Larsen, 2008). Before this point, “algebra” may

have been a synonym for the sorts of manipulations one makes to transform equations

into other equations that happen to be more useful for one’s purposes. An example

might be factoring equations in order to solve for the unknown. After this point,

“algebra” refers to structures like groups, rings, fields and modules; to standard

constructions like subgroup, quotient group, Cartesian product of groups; and most of

all, to proofs of all the important results, as opposed to calculations of answers.

Student difficulties with handling this transition are well-known, and researchers have

examined key concepts in these courses and how it is students understand them or fail

to understand them (Dubinsky, Dautermann, Leron, & Zazkis, 1994; Asiala, Dubinsky,

Mathews, Morics, & Oktac, 1997). More recently, instructional sequences have been

designed that guide students toward reinvention of some of these key concepts, namely

that of group, isomorphism, and quotient group (Larsen 2013; Larsen & Lockwood,

2013). We concentrate here on the concept of well-definedness, which plays a critical

role in the study of group theory.


Mathematician and philosopher Châtelet (1999) highlights the following features of

mathematical gestures. They have the power to awaken other gestures in their wake.

They are not just a movement in space; rather they bring into being a new manner of

movement. They are not decided in advance and then carried out; “one is infused with

the gesture before knowing it” (p. 10). By contrast with two other mathematical

devices, “functions” and “acts”, where a function can only express how to go from one

state to another, and an act, when finished, has no more to offer, a gesture is “elastic, it

can crouch on itself, leap beyond itself and reverberate” (p. 10). Perhaps most

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importantly for Châtelet, a gesture can “store up all the allusion’s provocative

virtualities, without debasing it into abbreviation” (p. 10). In turn, the diagram, which

“can transfix a gesture, bring it to rest, long before it curls up into a sign” (p. 10), can

go on giving and evoking and suggesting without emptying itself.

DATA COLLECTION

35 lectures in a third-year undergraduate mathematics course (Groups) at a mid-sized

University in Western Canada were video-recorded. The video camera was focussed

exclusively on the lecturer “J” (and not the students). A transcript was made of all of

the video-recordings.

DATA ANALYSIS

Well-defined: Episode 1

J begins writing the statement of the Orbit-Stabilizer Theorem (Theorem 7.3 of the

Gallian’s Contemporary Abstract Algebra): Let be a finite group of permutations of

a set . Then for each . In words this says that the

number of elements of (the order of ) is equal to the product of two numbers. The

first number counts how many different elements of the set are reachable when all

the permutations in are each applied to the element in . The second number

counts the number of permutations in which, when applied to the element in ,

give the result (in other words, these permutations ‘fix’ ). The professor begins the

proof with what he says is the “key idea” right at the start. He writes: “Let . We

claim that there is a 1-1 correspondence from the left cosets of in to the

elements of ” and then writes out the short argument that shows that the

Orbit-Stabilizer Theorem follows from this claim by a use of Lagrange’s theorem.

J then says: “There’s only one thing that requires any thought in this proof. I mean

once we have that, which is not entirely obvious, once we have that, the entire proof is

like turning the handle.” It will turn out that the “one place where we need to think” is

to show that a certain map is well-defined.

He proceeds to define a map as follows: . He demonstrates

how obvious this choice is by asking the class for what the right hand side ought to be,

and gets the answer from a student, commenting “what else could it map into”.

Then he asks them to leave three lines blank, and leaving this space blank on the board

himself, he titles two lines “ is 1 – 1” and “ is onto”. He says that the 1 -1 part will

be a one-liner in a moment, and then proves that the mapping they’ve defined is

onto. Once finished, he returns to the empty space, saying “What’s this missing proof-

missing part of the proof doing? I mean what- what else can- why don’t we just show

it’s one one and onto and go home”. A student answers: “show that it’s like

well-defined”:

For the following lines I will also keep track of the hand movements he makes (any

hand movements will be enclosed in curly parentheses):

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We said well just pick a coset {touches the term phi stab G of i} phi stab G of i {holds out

hands, palms up, shaking them slightly}. And the problem is {hands are still out, together,

not moving} let’s suppose that I picked- {touches the term phi stab G of i again} there are

different ways of calling the same coset right? {returns to his earlier pose, hands together,

palms up, holding an imaginary coset} You can have different coset reps, {hands move

apart slightly} I can have some psi that lives inside the coset, {writes ‘psi’ in the air with

his index finger, then points to the phi stab G of i term on the board} I could have some psi

stab G of i {holds out hands, palms up} that could mean the same coset {the first half of the

key gesture WD1: the hands draw apart dramatically, separated by a foot}. But let’s

suppose that phi and psi send me somewhere different {the finale of the key gesture WD1:

he alternates moving each hand a little, weighing imaginary cosets, and then immediately

on ‘send’ both hands turn into index fingers that fly out towards the class, splayed each at

45 degrees to the forward direction}.

Here, in the hands, are all the reverberations of ‘well-definedness’. The one object,

that is really two, and might therefore go to two different places; they must show it

really goes only to one place. The function, that they have defined as taking an input

and giving an output, hides almost as a secret that it is taking an input with a special

name out of a bag of objects each with different names. They have decided to consider

all objects in the bag as being the same. In the hesitation in his hands between the

hands being together, and the hands being separated, is the hesitation of even realizing

that they must check that their definition “makes sense”.

J turns to the board in order to write what he has gestured:

So let’s see what the picture- what’s the problem. The problem is that we start off here

{draws a circle right after “problem”} and let’s suppose we have two different names for

the same coset {writes out the equation phi stab G of i = psi stab G of i}. Right, that’s the

same coset with two different coset representatives. OK and now T tells me - well in one

case {puts dot in the middle of the circle} if you use the first name {draws line emerging

from center, touches left hand side of equation, adds arrow to line, labels line with T} I’m

gonna map you to phi of i {writes phi of I at the end of this labelled line}. But if you use

the second name for the same object {draws second line emerging from center, slanted

down and to the right; adds arrow to line, holds right hand side of equation} I’m gonna map

you to psi of i {labels line with T, writes psi of I at the end lf this labelled line}. That would

not be good {walks away from the board}. OK so actually what we need to show {walks

back to diagram} is that if this happens {uses index and pinky finger to simultaneously

hold both sides of the equation} these are the same {carries his gesture over to the ends of

the two labelled lines, and holds simultaneously phi of i and psi of i}.

Figure 1: Well-definedness diagram in the Orbit-Stabilizer Theorem

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Note that the source here is one circle, with the two names given as a left hand side and

right hand side of an equation, whereas with the earlier gesture WD1, the hands which

had once been together, diverged in a movement that he can’t quite replicate on the

board. The “two-ness” of the source, and how it is related to the “one-ness”, is more

direct and vital in WD1. Note too that the diagram is incomplete; it requires the

powerful finishing gesture of holding the equality and then holding the two separate

endpoints of the arrows to meld them into one unique outcome. However, WD1 was

also incomplete, as the two fingers splayed out in two different directions, and they

really ought to wind up at the same place. Now he is ready to finish the proof:

So what we need to show is that T is well-defined {writes “T is well-defined”}. And by all

that discussion {waves his index finger over the diagram and the equation above it} we just

need to show that this implies this {touches equation on first “this”; touches in between the

ends of the two arrows on second “this”; realizes this is not what he means} sorry, this

implies that these are equal {touches equation on “this”; holds both ends of the two arros

with index finger and pinky finger simultaneously}. And what I’m gonna say there {points

to the line in his proof that says “T is 1-1”, maintains the double finger hold} is reverse the

above argument {still holding}. Because one to one says that if these are equal then these

are equal ok {lifts index and pinky at “if”, retouches with them at first “these”, carries the

gesture over to the equation and simultaneously holds both sides on the second “these”}.

So as promised earlier, proving that T is 1-1 is indeed a one-liner. We see now in this

situation that the concepts of injectivity and well-definedness are very close to one

another; reversing the gesture of one gives you the gesture for the other. The gestures

have evoked each other.

Well-defined: Episode 2

WD2 occurs just after his proof of Theorem 24.1, which states that for any element

of a finite group , the order of the conjugacy class of is equal to the index of the

centralizer of in . Again he has defined a map T (from the set of left cosets of the

centralizer of to the conjugacy class of ), and now that we are near the end of the

course, he simply asks them to “check” that T is well-defined, 1-1 and onto, from

which the theorem follows. In his post-proof comment phase, he observes:

Well-defined is if you like, the converse of one one. Well-defined is if you take two things

with the same name {puts notes down, holds both hands up in the air in front of him and

above his head} and then you apply the map {holds hands there, shakes them at “apply the

map”} you end up in the same place {hands dive down, end up clasped together at waist

level, see Figure 2}. So two things with the s- {repeats: hands back up in the air in the

earlier front and high position, shaking them} with the same name end up equal to each

other {shakes them, hands dive down to lower clasped position}. One one is the images

are equal to each other {looks down at this hands, shakes them a little} the things that you

started off with are equal to each other {hands go up, separate, eyes track them, then he

brings the hands horizontally together until they are joined, see Figure 3}. So if you like

{one hand at the bottom, one hand sweeping the space up and down between the lower

position and the upper position} one is the reverse process of the other OK.

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Figure 2: Well-definedness gesture WD2, bottom position. Figure 3: Well-defined

gesture WD2, top position.

WD2 is the most complete well-definedness gesture in the course, and although I have

named it “well-definedness”, it includes in a natural way a 1 – 1 gesture.

When must we check for well-definedness?

It is important at this point to discuss the “function” gesture. J sometimes indicates the

result of performing a mapping using his hands and fingers. There’s a place to start

(the input), a little shaking or movement to activate it, a clear line or direction of

movement, and then a final resting place. It is exactly this gesture that the

well-definedness gestures must both approach (because if a function is indeed

well-defined, then the “function” gesture is the one that is appropriate!), and also avoid

(because if a function were obviously well-defined, we would not have any skepticism

or doubt, and there would be nothing to check!). This is an interesting site, because it is

not often that students are in a position in an undergraduate mathematics class where

they are asked to do something, but aren’t given an explicit prescription of exactly

when to do the thing asked. The issue comes up on yet another occasion:

1 S: How do you see that?

2 J: Ah, how do you see that it needs to be well-defined.

3 S: I understand that things need to be well-defined-

4 S: When you’re looking at something for the first time?

5 J: That’s why I’m giving it as an example.

Perhaps the student wants to know why on so many occasions mappings are defined

and they don’t bother to check that they are well-defined, and on other occasions it

becomes an issue. What is the deciding factor that makes it an issue? J’s next attempt

at a response is to say that the “politically correct” answer is that “you should always

check that a mapping is well-defined as a matter of principle” while immediately

acknowledging that “most of the time you just don’t do it because it’s kind of just

tedious and most mappings are”. The student presses further, asking “so should we do

that every time?”, to which J answers:

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In this- like I said, I’ve drawn particular attention- the coset stuff- when you’ve different

ways of naming the same object you gotta be careful within this chapter on

homomorphisms we’re gonna see this idea several times. So just sort of heighten your

sense of- your awareness or sense of danger around this.

There is a lot of experience in play in each of J’s lines. And yet we might feel with the

student that a clear-cut answer is being avoided.

I suspect that a clear-cut answer is impossible for a few reasons. Let’s first cite a

standard description of failing to be well-defined:

A proposed function can fail to be well-defined in two ways: (1) some in the

domain can fail to have a in the co-domain to which to map or (2) some in the

domain can be mapped (ambiguously) to two different ’s in the co-domain. (Hunter, 94).

The first way of failing to be well-defined occurred once in the course. It is the second

way that occurs most frequently in group theory. So this might be a reason why J

couldn’t give a clear-cut answer in the time he had available: he would first need to

explain the two sorts of ways functions can fail to be well-defined, and also begin to

give examples of the first way because it has less commonly been seen by them. A

second reason is that it is very hard to enumerate all the ways in which you might be

faced with a situation where (2) applies. So J does the best he can: he mentions the

issue every time it comes up, and he solves examples where the issue emerges so the

students can see it come up in the sorts of circumstances that it does.

The student’s repeated questioning, J’s honest and careful replies: the moment is

significant. It is perhaps the clearest moment in the course where there remains some

irreducible gap between student and teacher that might be bridged only by hands-on

experience.

Well-definedness and quotient groups

I turn now to the most critical appearance of this concept in introductory group theory.

The context is Theorem 9.2 (tagline “Factor Groups”). This is the theorem that gives a

necessary and sufficient condition for a subgroup of a group to be normal in :

is normal in if and only if the set of left cosets forms a group under the

operation . Before stating the theorem J illustrated using the dihedral

group that such a binary operation of multiplication need not be well-defined. Just

before he begins the proof he comments:

And what we’ll see as we do the proof right now, we’re gonna see that the only way that

this operation could fail to be a group is if the operation is not well-defined. Everything

else is gonna work just fine. We’re gonna get all the properties we would want from a

group formally, but the only place that can fail is not well-defined.

J stressed “only” and “fail” in a pronounced manner. J makes clear here, (and will

often repeat this important observation), that the only obstruction to having a group

structure on the set of left cosets is that the multiplication might not be well-defined.

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J now tackles the question at hand. Note that he begins in a very similar way to how he

began just before he drew the diagram in Figure 1, asking “what’s the problem” (recall

that recognizing that the problem exists in the first place was exactly the student’s

question in the last section):

So what’s the problem. The problem- ok let’s- let’s say that a H is a prime H {writes the

equation a H = a prime H}. We have two different names for the same coset here and we

have two different names for another coset {writes “and b H = b prime H”}. Now what

could be the problem. Well if this is a H {draws a circle with label a H = a prime H} and

this is b H {draws circle below the first one, with label b H = b prime H} then if we form-

if we use these two and we get to ab H {draws arrows that end at a new circle, labels it ab

H}, by combining them like - we don’t want to get to a different place if we used different

names a prime b prime H {draws arrows that end at a new circle, labels it a prime b prime

H}. So the question is, can that happen, and we hope not, OK? {draws a question mark,

points with his pen to his entire diagram}.

So he explicitly reminds them of the diagram in Figure 1. The difference here is that

we have yet another doubling in play. Let’s get the doublings straight. The first

doubling was that of the source object: two names for the same thing. The second

doubling was the output object: conceivably our mapping could take us to two places,

hopefully it won’t. The third doubling here is new, and is due to the fact that our

proposed binary operation here is a map from to . His finished

diagram is in Figure 4:

Figure 4: Well-definedness and the Quotient Group Theorem

This is a bit like an image of two eyes that are experiencing double vision, instead of

focussing correctly on a single object, which is not unlike the experience of dealing

with a well-definedness situation. Like Figure 1, this diagram is incomplete, but this

one announces it, in the form of a question mark. Like Figure 1, the input is not

doubled, but labels above each input express the doubling. Finally, like Figure 1, the

output is doubled.

DISCUSSION

The quotient group is a central concept in group theory, and plays a role in virtually

every development from that moment forward. Since well-definedness of the

multiplication operation is the only hurdle that must be cleared in order for the set of

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cosets to become a group, we can see that well-definedness is a critical ingredient in

group theory. It is a crucial realization that a natural attempt at a definition of a

mapping or operation might fail to make sense. Negotiating this conceptual leap was a

major event in the early history of group theory, performed implicitly by Galois,

revealed in an increasingly explicit manner in the work of later mathematicians,

culminating in work by Holder (Nicholson, 1993).

Students must negotiate this conceptual leap as well. We have studied here the

approach of one lecturer to sharing with his students how he thinks about the

well-definedness of mappings. With the movement of his hands he clarifies the

connection to what it means for a function to be one to one. He also illuminates the

sorts of situations in which one will have to argue that a mapping is well-defined: we

must have two objects with different names that we are considering to be the same. For

this lecturer, explaining well-definedness – what it is, and when to expect to have to

prove it – requires hand gestures, and diagrams born of those gestures.

References

Asiala, M., Dubinsky, E., Mathews, D. M., Morics, S., & Oktac, A. (1997).

Development of students' understanding of cosets, normality, and quotient

groups. The Journal of Mathematical Behavior, 16(3), 241-309.

Châtelet, G. (2000). Figuring space: philosophy, mathematics, and physics. Kluwer

Academic Pub.

Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning

fundamental concepts of group theory. Educational studies in Mathematics, 27(3),

267-305.

Hunter, D. (2010). Essentials of discrete mathematics. Jones & Bartlett Publishers.

Nicholson, J. (1993). The development and understanding of the concept of quotient

group. Historia Mathematica, 20(1), 68-88.

Larsen, S. P. (2013). A local instructional theory for the guided reinvention of the

group and isomorphism concepts. The Journal of Mathematical Behavior, 32(4),

712-725.

Larsen, S., & Lockwood, E. (2013). A local instructional theory for the guided

reinvention of the quotient group concept. The Journal of Mathematical

Behavior, 32(4), 726-742.

Weber, K & Larsen, S. (2008). Teaching and learning group theory. Making the

Connection: Research and Teaching in Undergraduate Mathematics Education,

(73), 139.

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NEGOTIATING MEANING: A CASE OF TEACHERS DISCUSSING

MATHEMATICAL ABSTRACTION IN THE BLOGOSPHERE

Judy Larsen


Many mathematics teachers engage in the practice of blogging. Although they are

separated geographically, they are able to discuss teaching-related issues. In an effort

to better understand the nature of these discussions, this paper presents an analysis of

one particular episode of such a discussion. Wenger’s theoretical framework of

communities of practice informs the analysis by providing a tool to explain the

negotiation of meaning in the episode. Results indicate that the blogging medium

supports continuity of discussions and can allow for the negotiation of meaning, but

that a more nuanced treatment of the construct is necessary.

INTRODUCTION

I’ve been throwing around the term “pyramid of abstraction” recently, and there was

some great pushback on Twitter this evening. This post is my attempt to clarify what I

mean, and why I think it is a useful perspective to building students’ knowledge.

(Kane, 2015c, para. 1)

The blog post quote above was written by Dylan Kane, a mathematics teacher, after a

series of interactions on Twitter with Dan Meyer and Michael Pershan. The series of

interactions were prompted by Dylan’s original blog post that he shared on Twitter, in

which he explained his ideas about a metaphor for mathematical abstraction. The

discussion that resulted from this formulation presents an instance of negotiating the

meaning of a metaphor that may be useful in the teaching and learning of mathematics.

More importantly, it provides an example of how mathematics teachers are engaging in

the practice of blogging, which is the focus of this paper.

Mathematics teachers around the world are choosing out of their own will to create,

organize, and manage their own personal blog pages. On these publicly visible virtual

pages, they make relatively regular posts, which are most often related to their work as

mathematics teachers. These posts can include written and/or media enhanced recounts

from their teaching experiences, links to interesting resources, and responses to posts

that other bloggers have made.

However, since blog pages are individually managed and no universal blogging

platform exists, finding like-minded bloggers may at times be difficult. To mitigate

this issue, many bloggers extend their practice through Twitter, a universal

micro-blogging platform that allows users to create searchable profiles. There are now

361 profile listings on the Math Twitter Blogosphere directory (MTBoS, 2015), most

of whom have both a Twitter handle and a blog page listed on their profile. Although

micro-blogging on Twitter is slightly different in nature than blogging due to the 140

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character limit on each post, which forces users to communicate ideas more concisely,

it is still considered a form of blogging (Ebner, 2013). Blog pages allow users to

explore ideas deeply, but Twitter makes it easier for like-minded bloggers to connect.

Mathematics teacher bloggers often use both mediums to express ideas, linking

between the two when necessary. For instance, some bloggers include snips of Twitter

conversations in their blog posts, and other bloggers link to their blog posts within their

Twitter posts. In this way, teachers are communicating with peers across the world.

This sort of collegial interaction is unusual for teachers because teaching is generally

an isolated profession (Flinders, 1988). Mathematics teachers’ typical contact with

other mathematics teachers is limited to that of department meetings and teacher

lunchrooms, where teachers usually plan lessons and grade papers individually

(Arbaugh, 2003). Professional development initiatives require time, funding, and

facilitation, and are therefore limited to sparse one-time professional development

workshops throughout the year whose benefits are not generally found to be effective

in stimulating teacher collaboration (Ball, 2002).

However, there are over 600 mathematics teacher bloggers (Meyer, 2014) who seem to

be overcoming these constraints. It is evident that they spend hours writing publicly

about their daily practice, posting resources, and sharing their dilemmas with no

compensation and no mandate. This unprompted, unfunded, and unevaluated teacher

activity is a rich phenomenon of interest that deserves attention. Ironically, this

phenomenon is largely unstudied. Empirical investigations related to blogs in

mathematics education are limited to studies on the utility of blogging as a pedagogical

tool within either a mathematics course (Nehme, 2011) or a mathematics education

course (Silverman & Clay, 2010; Stein, 2009). These studies do not account for the

autonomous and self-driven nature of the blogosphere and there is no clear work in

mathematics education exploring the activities of these teachers.

As such, this study is guided by the overall research question of what the mathematics

teacher blogosphere occasions for teachers who engage in blogging in relation to their

practice. To this end, I pursue an investigation of one episode that exemplifies the type

of interaction that is possible within the mathematics teacher blogosphere.


Since blogging is an individual practice that is made public (Efimova, 2009), a

mid-level theory that considers situated participation is desirable. Communities of

practice (Wenger, 1998) is one such theory: it is a social theory of learning where

learning is considered as increasing participation in the pursuit of valued enterprises

that are meaningful in a particular social context. Practice is at the heart of Wenger’s

(1998) communities of practice, and a key aspect of practice is the ability to motivate

the social production of meaning. The continuous production of meaning is termed as

the negotiation of meaning, and is further defined by the duality between participation

and reification (Wenger, 1998). Participation is “the process of taking part and also to

the relations with others that reflect this process” and reification is “the process of

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giving form to our experience by producing objects that congeal this experience into

‘thingness’” (Wenger, 1998, p. 58). Both participation and reification shape the

participant and the community in which they participate in an ongoing manner.

Wenger (1998) claims that participation and reification imply each other, require and

enable each other, and interact with each other. Wenger (1998) notes that the benefit to

viewing the negotiation of meaning as a dual process is that it allows one to question

how the production of meaning is distributed. He notes that there is “a unity in their

duality, [because] to understand one, it is necessary to understand the other, [and] to

enable one, it is necessary to enable the other” (Wenger, 1998, p. 62). Various

combinations of the two will produce different experiences of meaning, and together

they can create dynamism and richness in meaning if a particular balance is struck.

Wenger (1998) notes that “when too much reliance is placed on one at the expense of

the other, the continuity of meaning is likely to become problematic in practice” (p.

65). If participation dominates, and “most of what matters is left unreified, then there

may not be enough material to anchor the specificities of coordination and to uncover

diverging assumptions” (p. 65). However, if reification dominates, and “everything is

reified, but with little opportunity for shared experience and interactive negotiation,

then there may not be enough overlap in participation to recover a coordinated,

relevant, or generative meaning” (p. 65). In essence, participation allows for

re-negotiation of meaning, and reification creates the conditions for new meanings. In

this interwoven relationship, the two aspects work together to drive the process of

negotiation of meaning. As such, this framework is suitable to examine the social

production of meaning within the mathematics teacher blogosphere with the aim of

determining what the blogosphere occasions for mathematics teachers.

METHOD

In the process of acquiring data from a larger study within the blogosphere, I have

acquired a notable list of mathematics teachers I now subscribe to. Every user will see

different posts based on who they subscribe to, and it is important to note that my

pervasively subjective position influences what I can notice in this ultra-personalized

and dense virtual environment. Aside from creating a profile, my activity has been

predominantly that of a ‘lurker’ in that I have not made significant contributions to the

blogosphere. Although the position of ‘lurking’ has been valuable, this position

changed slightly after my attendance to the MTBoS conference ‘Twitter Math Camp

2015’ (TMC15). After the conference, I found myself connected to this group of

MTBoS bloggers more than ever before. This shift in position allowed me to be closer

to the active conversations these members were having on Twitter.

It was during this time after my return from TMC15 that I encountered a particular

conversation that I flagged as interesting based on my theoretical framework. This

particular conversation was chosen for its power to illustrate a possible mode of

interaction in the blogosphere. After the conversation took place, I used storify.com to

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chronologically reconstruct it by identifying all tweets related to the conversation from

the feeds of each of the members who participated, arranging them chronologically.

I then contacted the initiator of the discussion, Dylan Kane, for an interview. I

interviewed him by proxy about his experiences in the blogosphere as well as about his

reactions to the conversation that had occurred on Twitter in relation to his blog posts.

The interview was voice recorded and transcribed completely. The transcription of the

interview with Dylan, the reconstructed Twitter conversation, and the blog posts Dylan

linked to within his Twitter posts comprise the data set for the analysis of this case.

Each of these sources of data were then coded according to notions of Wenger’s (1998)

participation and reification. The data were then arranged into a chronological order so

that a complete discussion could be seen with codes indicating moments of

participation and reification. In this case, participation was considered to be any

action that a blogger took as part of the blogging practice, and reification was

considered to be any trace that was left from a participation. The interplay between

these aspects was then considered in relation to the data, and conclusions were drawn

about the nature of the negotiation of meaning as exemplified in this case of

mathematics teacher blogging. For purposes of brevity, a reduced version of the

analysis is presented in what follows.

ANALYSIS

On June 25, 2015, Dylan participates in thinking about his past formulations about

problem solving. He participates in writing this idea out in a blog post (Kane, 2015b).

Within the body of this post, Dylan discusses his ideas on how teachers can help

students deal with solving difficult mathematical problems by helping them become

better equipped to transfer prior knowledge to new contexts in mathematics. At one

point, he brings up the idea of a ‘ladder of abstraction’ that he has heard references to.

He explains that the ‘ladder of abstraction’ refers to “the idea that students’

understanding begins with the concrete, and climbs a metaphorical ladder as it

becomes more and more abstract” (Kane, 2015b, para. 11). However, he believes that

the metaphor is incomplete.

Based on this perspective, I think the metaphor of a ladder of abstraction would be better

replaced by a pyramid of abstraction . . . I worry that the ladder of abstraction metaphor

leads me to believe that, once a student understands one concrete example of a function,

they are ready for a more abstract example. While some students may be, I want to focus

on building a broad base first, and then moving up the pyramid after we have spent time

analyzing the connections between the examples and the underlying structure.

(Kane, 2015b, para. 12-13)

Although this formulation of the ‘pyramid of abstraction’ is built on a history of

participation and reification that Dylan has engaged in, I take this to be the beginning

of this particular episode of negotiation of meaning. By writing this down and

publishing it as a blog post that is publicly visible, Dylan has created a reification, and

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it is likely that this reification is influenced by the very process of participating in

writing the blog post.

A month later, on July 17, 2015, Dylan participates in writing a reflection about his

experiences at PCMI in relation to mathematical problem solving. Within this

reflection, he makes a reference back to his June 25, 2015 post regarding his

description of a ‘pyramid of abstraction.’

This gets at something I wrote about recently that I called the pyramid of abstraction – that

students build abstract ideas from looking at connections between a wide variety of

examples, rather than simply jumping from concrete to abstract. (Kane, 2015a, para. 6)

After participating in this reflective moment, trying to recapture his experiences from

a professional development encounter, and connecting his experiences with something

he had previously written about, he writes his ideas down and publishes them. In this

act, he produces another reification. This reification now has a rich history, and

through the use of hyperlinks, one can trace the ideas back to where they evolved from

for Dylan. It not only encapsulates the history of ideas, but it also prompts further

participation for either Dylan or other mathematics teacher bloggers who participate

in reading his post.

Simultaneously to posting this blog post, Dylan also publishes a link to it on Twitter.

Doing this creates a rich reification because of the number of traces that it leaves and

the visibility it affords. It is also simultaneously a manner of participating in the

blogging practice. Wenger (1998) notes that participation and reification are so closely

tied that sometimes they may be difficult to pry apart. This is evident here, particularly

due to the nature of the blogging practice.

Shortly after publishing the link to his post on Twitter, Michael Pershan responds in

agreement with Dylan’s reference to the ‘pyramid of abstraction’ metaphor (Figure 1).

Figure 1 Michael Pershan's response to Dylan's post

In the interview with Dylan, he confides that he and Michael have discussed the idea of

a ‘pyramid of abstraction’ for a while and that that they have come to a shared meaning

for the term. Hence, this idea of a ‘pyramid of abstraction,’ which is itself a reification,

has a long history of negotiation of meaning.

Fifteen days after Dylan’s Twitter post of the link to the blog post that included a link

to his ‘pyramid of abstraction’ post, Dan Meyer poses a question to Dylan and Michael

on Twitter about the meaning of the apex of the ‘pyramid of abstraction’ (Figure 2).

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Figure 2 Dan Meyer's question about the pyramid of abstraction

The power of this reification is that it is pervasively public. Dan Meyer currently has

39, 320 followers. Michael Pershan has 3,423 followers, and Dylan Kane has 785

followers. The nature of Twitter is that if a person tags another person in a post, only

people who follow both of those people see the post. Since Dan, Michael, and Dylan

are all mathematics teacher bloggers and share followers, the number of people who

see such a post is large. Hence, Dan’s reification is bound to prompt further

participation, leading to the continuous evolution of the negotiation of meaning around

this particular notion of abstraction within mathematical problem solving.

Over the course of a few hours, these iterations of participation and reification

continue, and the negotiation of meaning evolves the notion of ‘abstraction’ and its

role in mathematical problem solving. Initially, Dylan and Michael restate their

interpretations of the meaning of a ‘pyramid of abstraction.’ To this, Dan responds

with a critique claiming that there can be no ‘top’ to abstraction. This prompts a

re-negotiation of the term ‘abstraction,’ and prompts Dylan to re-define what he means

by the concept that he has reified (Figure 3).

Figure 3 Negotiation of the term 'abstract'

In this interplay of multiple iterations of participation and reification, the meaning of

the term ‘abstraction’ is negotiated, and it is clear that nuances within their

interpretations exist. In this interaction, they are realizing that the metaphor can mean

different things to other mathematics teachers. In fact, when Dylan reflects on this

experience in the interview, he states that the biggest thing he has learned from this

negotiation, is that metaphors can mean different things to different people.

After a few hours, Dylan writes a blog post in response to the discussion and links to it

on Twitter. In this post, he restates his interpretation of ‘abstraction’ by claiming that

he has used the term to refer to something that helps students build “a concept in a way

that they can apply it in a variety of different contexts” (Kane, 2015c, para. 6).

However, he also expresses an acknowledgement that this may not be how others view

‘abstraction.’

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Do I Actually Understand Abstraction? I’m redefining abstraction a bit, and I’m also

lumping knowledge transfer as one giant idea, which it might not be.

(Kane, 2015c, para. 19)

This blog post may be seen as a reification of a reflective form of participation.

However, it seems to represent a different form of reification than that merely of a

trace left from a participation in the blogging practice. This reification seems to

indicate a heightened level of awareness to which Dylan has arrived. However, the

construct of reification as defined by Wenger (1998) does not allow for this

differentiation.


In the analysis, it is found that participation in the mathematics teacher blogging

practice includes reading, thinking about, and writing blog or micro-blog (Twitter)

posts in relation to mathematics teaching and learning. In turn, reification in this

practice is that of publishing such posts. It is clear that within the mathematics teacher

blogging practice, participation and reification are intimately intertwined, and their

close co-evolving and co-implicated relationship drives the negotiation of meaning

among participants and ensures continuity of the negotiation. This points to the

powerful nature of the medium itself.

This analysis also reveals a need to redefine participation and reification for

mathematics teacher blogging in particular because various forms of each have

surfaced. As noted above, various forms of participation exist, and each form implies a

different mode of interaction in the negotiation of meaning. In particular, some

instances of participation are prolonged and accompanied by reflective activity, which

contrasts the instances where a quick knee-jerk tweet is made. An exploration of the

role of reflection in blogger participation may be fruitful in future studies.

Further, it is also evident from the analysis that certain reifications can be of higher

order than others. In the practice of mathematics teacher blogging, each post forms a

reification because it leaves a trace of a participation. However, certain reifications

indicate that a shift in meaning has occurred. There is currently no terminology within

Wenger’s (1998) frame to refer to such occurrences. However, it may very well be that

a new term needs to be introduced to refer to higher order reifications.

Finally, it should be noted that the construct of negotiation of meaning also has the

power to reveal what is being negotiated. In this case, the teachers found it valuable to

invest time into engaging in a prolonged episode of negotiating the meaning of

‘abstraction.’ Looking for more of these instances may help identify teaching-related

issues that mathematics teachers take interest in. Most evidently, the blogging medium

can make it possible to trace the development of ideas in the blogosphere. Each idea

has a history and a future. The history is traceable, and the future is quickly unfolding.

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References

Arbaugh, F. (2003). Study groups as a form of professional development for secondary

mathematics teachers. Journal of Mathematics Teacher Education, (February), 139–163.

Ball, D. (2002). What do we believe about teacher learning and how can we learn with and

from our beliefs? In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, &

K. Nooney (Eds.), Proceedings of the 24th Annual Meeting of the North American Chapter

of the International Group for the Psychology of Mathematics (pp. 2–18). Athens, GA.

Ebner, M. (2013). The influence of Twitter on the academic environment. In B. Patrut, M.

Patrut, & C. Cmeciu (Eds.), Social Media and the New Academic Environment:

Pedagogical Challenges (pp. 293–307). Hershey, PA: IGI Global.

Efimova, L. (2009). Passion at work: Blogging practices of knowledge workers. Novay PhD

Research Series (Vol. 024).

Flinders, D. (1988). Teacher isolation and the new reform. Journal of Curriculum and

Supervision, 4(1), 17–29.

Kane, D. (2015a). More insights from Dylan Wiliam. Retrieved from

https://fivetwelvethirteen.wordpress.com/2015/07/17/more-insights-from-dylan-wiliam

Kane, D. (2015b). Problem solving: Classroom application. Retrieved from

https://fivetwelvethirteen.wordpress.com/2015/06/25/problem-solving-classroom-applica

tion/

Kane, D. (2015c). Pyramid of abstraction. Retrieved from

https://fivetwelvethirteen.wordpress.com/2015/07/31/pyramid-of-abstraction/

Meyer, D. (2014). The 2014 TMC sessions: Meet your MTBOS. Retrieved July 16, 2015,

from http://tmc14.mrmeyer.com/2014/07/22/meet-your-mtbos/

MTBoS. (2015). MTBoS Directory. Retrieved July 16, 2015, from

https://sites.google.com/site/mtbosdirectory/

Nehme, Z. (2011). Blogging and the learning of linear algebra concepts through contextual

mathematics. Mathematics Teaching, 57, 1–11.

Silverman, J., & Clay, E. L. (2010). Online asynchronous collaboration in mathematics

teacher education and the development of mathematical knowledge for teaching. The

Teacher Educator, 45(1), 54–73.

Stein, C. (2009). Using blogs to support mathematics instruction among first year elementary

school teachers. Society for Information Technology & Teacher Education International

Conference, 2977–2981.

Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. New York,

NY: Cambridge University Press.

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DISCURSIVE PRACTICES USED IN DEFINING

MATHEMATICAL LEARNING DISABILITIES: A BRIEF

TEXTBOOK ANALYSIS

Peter Lee


This paper compares two separate descriptions of mathematical learning disabilities

given in a textbook on learning disabilities. One description is from an early edition of

the textbook and the other description is from a later edition of the same textbook.

Critical discourse analysis is applied to both descriptions to investigate the changing

discursive practices used in defining mathematical learning disabilities. While a

“medical discourse” dominates both the early and later editions, there is evidence of

other competing discourses which suggests some instability around the boundaries of

what constitutes a “mathematical learning disability.”

INTRODUCTION

There has been much research on reading disabilities (e.g. dyslexia) but much less

research done in the area of mathematical learning disabilities (MLD) (e.g.

dyscalculia). Yet the prevalence rates for dyslexia and dyscalculia are approximately

equal (around 5% to 8%). Moreover, much of the research on MLD is from a cognitive

psychology and special education perspective and more research needs to be done from

a mathematics education perspective. The recent growth in the use of sociopolitical

perspectives in mathematics education has introduced new analytic tools that were not

traditionally used in the field such as discourse analysis. This research report will

apply Norman Fairclough’s (1992) version of critical discourse analysis (CDA) in

order to analyze some of the changing discourse practices used in the definition of

MLD found in a well-established textbook on learning disabilities.


The theoretical framework I have chosen for this analysis is borrowed from Gina M.

Borgioli’s (2008) article examining the practices around the identification and labeling

of students with learning disabilities in mathematics. Her framework applies a critical

pedagogy (Kincheloe & McLaren, 1994) perspective looking through the lens of

ableism (Rauscher & McClintock, 1997). Critical pedagogy “seeks to uncover

hegemonic (power) relations, ideologies, and inequities in education, critique

instrumental rationality, and inspire a movement for change toward social justice”

(Borgioli, 2008, p. 132). One key assumption of critical pedagogy relevant to this

paper is “acknowledging schooling as a form of cultural politics that endorses only

particular forms of knowledge, thus creating a dominant group of successful knowers

and ‘others’” (Borgioli, 2008, p. 132). Moreover, Borgioli (2008) interprets ableism to

be

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A deliberate act of exclusion and discrimination. It entails those in a position of (political,

economic, and/or educational) power narrowly defining what society and educators are to

consider as acceptable ‘school mathematics’ as well as acceptable evidence for students’

demonstration of proficiency in school mathematics. (p. 133)

METHODOLOGY AND METHODS

Norman Fairclough’s (1992) CDA is a useful framework for analyzing discourse as

social practice and, in particular, the discursive practices used in defining MLD within

textbooks. The main objective of Fairclough’s approach to language analysis is to

study social and cultural change. Shifts in language use play a central role in the

understanding of changes in social phenomena. Fairclough’s CDA synthesizes two

different senses of discourse—the social-theoretical sense (such as Foucault’s) and the

“text-and-interaction” sense—and forms a three-dimensional model in the following

way:

Any discursive “event” (i.e. any instance of discourse) is seen as being simultaneously a

piece of text, an instance of discursive practice, and an instance of social practice. The

“text” dimension attends to language analysis of texts. The “discursive practice”

dimension, like “interaction” in the “text-and-interaction” view of discourse, specifies the

nature of the processes of text production and interpretation, for example which types of

discourse (including “discourses” in the more social-theoretical sense) are drawn upon and

how they are combined. The “social practice” dimension attends to issues of concern in

social analysis such as the institutional and organizational circumstances of the discursive

event and how that shapes the nature of the discursive practice, and the

constitutive/constructive effects of discourse referred to above. (p. 4)

This synthesis of the socially and linguistically oriented views of discourse is what

Fairclough calls a “social theory of discourse.”

Fairclough’s (1992) multi-dimensional approach emphasizes the importance of text

and language analysis (such as systemic functional linguistics) in discourse analysis

and has developed an explicit and operational approach for researchers to analyze

discourses. There are two key focal points of any analysis in Fairclough’s CDA: the

order of discourse (the totality of discursive practices of an institution, and the

relationships between them) and the communicative event (an instance of language

use). Due to the small scope of this paper, the focus will be primarily on the

communicative event (the two textbook excerpts), and Fairclough’s three-dimensional

model of discourse (as text, discursive practice, and social practice) will be applied to

it. While my analysis of these three dimensions for each excerpt will be done together,

each dimension should be separated analytically. For example, the analysis of

discourse as text involves a focus on the linguistic features of a text (linguistic

analysis). This may involve an analysis of vocabulary, syntax, and the grammar of

sentences. The analysis of discourse as discursive practice concerns how authors draw

on existing discourses and genres to create a new text. A key question to ask is: Are

discourse types (genres and discourses) used conventionally or creatively?

Conventional discourse practice involves a normative use of discourse types and helps

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to reproduce the relationships in the order of discourse. On the other hand, creative

discourse practice often mixes together a number of genres and discourses and helps to

restructure the boundaries of the order of discourse. The last dimension to be

considered when analyzing a discourse sample is discourse as social practice: What is

the relation between this discourse practice and the larger social structure to which it

belongs (is it conventional and normative or creative and innovative)? Does it

transform or reproduce existing social practices? At this point, theories beyond the

discourse analytical type such as social or cultural (or math education) theory need to

be drawn upon.

ANALYSIS AND DISCUSSION

I have selected two short, but related excerpts from two textbooks on learning

disabilities for analysis using this three-dimensional approach. The excerpts are from

the same textbook but different editions (Lerner, 1989; Lerner & Johns, 2015) and

reproduced below. Janet W. Lerner is the primary author of each textbook and her

textbook is now in its thirteenth edition. I have selected this textbook because of its

longevity and it is representative of many other textbooks on learning disabilities.

Also, the overall structure and organization of the textbook, surprisingly, has not

changed much, making comparative analysis somewhat easier. The two excerpts I

have selected are from the fifth (1989) and thirteenth (2015) editions respectively.

Both textbooks have a common chapter related to MLD, and the section of this chapter

I have chosen to examine is from the introductory description and definition of MLD.

Analysis of excerpt from the 1989 edition

The first excerpt from the 1989 edition is entitled “Characteristics of Students with

Mathematics Disabilities” and is reproduced below:

The subject of remedial mathematics has received much less attention than remedial

reading. Yet for many youngsters with learning disabilities, mathematics is the major area

of difficulty. A severe mathematics learning disability and related conceptual disturbances

in learning quantitative elements are sometimes referred to as dyscalculia. As with the

term dyslexia, which is applied to a severe reading disability (see Chapter 12), dyscalculia

has medical connotation suggesting a central nervous system involvement. Whatever it is

called, severe difficulty in learning mathematics can be a debilitating problem in school

and in later life. Not all students with learning disabilities have difficulty with number

concepts. In fact, some students with severe reading disability have strong mathematics

skills. Yet mathematics disorders do affect a significant portion of the learning-disabled

population. McLeod and Crump (1978) found that about one-half of students with

learning disabilities require supplemental work in mathematics, although only 10 percent

were seriously deficient in mathematics.

Many of the symptoms of learning disabilities can be linked to mathematics difficulties.

Factors that play a role in mathematics learning include language, conceptual,

visual-spatial, and memory abilities (Johnson, 1987). Difficulties in spatial relationships,

visual perception, symbol recognition, language development, and cognitive learning

strategies all have obvious implications for quantitative learning. It must be remembered,

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however, that each student is unique, and not all students who have a mathematics problem

will exhibit the same characteristics. We need much more research on the subtypes of

mathematics disabilities (Keogh and Babbitt, 1986). (Lerner, 1989, pp. 430-1)

Firstly, the author does not clearly define what a mathematics disability is or even

clearly state what to call it. She draws on many different discourses in an attempt to

describe students who have difficulties with math, but no one discourse seems to be

dominant and fix the meaning of a mathematics disability (according to Fairclough,

these interdiscursive mixes can be a sign of instability). It seems so little is known

about mathematics disabilities that it needs to be juxtaposed with discourses on reading

disabilities in order to gain any meaning itself: “remedial mathematics has received

much less attention than remedial reading,” and “dyscalculia” is compared to

“dyslexia.”

There also appears to be inconsistencies in what to call or describe the very thing to be

defined. The phrase “mathematics disabilities” is used in the title of the section, but

appears only once in the body of the text at the very end as part of the phrase “subtypes

of mathematics disabilities.” The text begins by talking about “remedial

mathematics,” then the next sentence is about mathematics difficulties in the context of

“youngsters with learning disabilities.” The following sentence contains three

different labels: “severe mathematics learning disability,” “related conceptual

disturbances in learning quantitative elements,” and “dyscalculia.” We come closest to

the origins of a mathematics disability when dyscalculia is described as having “a

medical connotation suggesting a central nervous system involvement.” This

instability in the label used to describe a mathematics disability is reflected in the

modalities “sometimes referred to,” “suggesting,” and “whatever it is called”

indicating low affinity for the statements and descriptions made.

Although there may exist some uncertainty in the text around the label itself (the text

suggests several names: mathematics disability, mathematics learning disability,

dyscalculia, mathematics difficulties, and mathematics disorders), there is a strong

sense that mathematics disabilities have a biological basis because of the medical

discourse that dominates the text. For instance, dyscalculia “has a medical connotation

suggesting a central nervous system involvement.” There is other medical terminology

used throughout the text that suggests a medical discourse in relation to mathematics

disabilities reminiscent of a doctor remedying a patient’s illness: “remedial,” “severe,”

“debilitating,” “disorders,” “deficient,” “symptoms,” “factors,” and “subtypes.” This

medical discourse is reinforced by the impersonal and objective nature of the text as if

one were reading a medical book. The frequent use of passive clauses and

nominalizations aim to obscure agency and the attribution of responsibility as if

mathematical disabilities are a natural phenomenon. For example: “The subject of

remedial mathematics has received much less attention than remedial reading.”

Attention from whom? Teachers? Academics? Psychologists? Also, the author

writes: “A severe mathematics learning disability and related conceptual disturbances

in learning quantitative elements are sometimes referred to as dyscalculia.” Again,

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who is doing the referring? There are many more examples like these that obscure the

individuals doing the labeling (diagnosing?) and create a feeling of depersonalization

throughout the passage.

Despite the apparent objective nature of mathematics disabilities that the text suggests,

there are some hints of uncertainty evidenced by the theme “not all students” that

begins the sentence “Not all students with learning disabilities have difficulty with

number concepts” and the secondary clause “not all students who have a mathematics

problem will exhibit the same characteristics.” The qualifier “not all” suggests the

complexity of mathematics disabilities and that it is not homogeneous in terms of

causes or characteristics. Moreover, there is hedging suggested by the statements “It

must be remembered, however, that each student is unique” and “We need much more

research on the subtypes of mathematics disabilities.” This is consistent with research

that has been unable to isolate the cognitive variables that underlie MLD due to many

confounding factors. For instance, difficulties in math and reading acquisition are

often attributed to the same cognitive processes (phonological in nature). However,

what’s most notable about these hedges is the personalization of the author. Rather

than using the imperative statement “Remember that each student is unique,” she is

more passive and indirect: “It must be remembered, however.” The author seems to be

taking a softer tone here, one that’s less authoritative, and seems to hint at some

empathy towards the students by reminding readers of their uniqueness. The use of the

personal pronoun “we” is also quite telling. While somewhat ambiguous in meaning,

the “we” seems to be positioning the readers (likely teachers) and the author as equals

in the sense that “we” all share the same goal in better understanding the mathematics

disabilities of “our” students. This less objective and more “sympathetic” discourse

competes with the more dominant “clinical” discourse that runs throughout the text.

Analysis of excerpt from the 2015 edition

The second excerpt is entitled “Students With Mathematics Difficulties and Students

With Mathematics Learning Disabilities” and is reproduced below:

Many students have difficulty in acquiring and using mathematics skills. Researchers

differentiate 2 different groups: (1) students with math difficulties and (2) students with

mathematics learning disabilities. In this book, we offer strategies for teaching both

groups. Students with math difficulties perform poorly in mathematics achievement tests.

Over 30% of eighth-grade students score below basic math performance on the National

Assessment of Educational Progress (NAEP) (Maccini, et al., 2008; Mazzocco, 2007).

In contrast, students with mathematics learning disabilities comprise about 6% of the

general population (Mazzocco, 2007). Mathematics learning disabilities is a biologically

based disorder and is related to difficulties in cognitive processing and brain functioning.

Research with functional Magnetic Resonance Imaging (fMRI) shows these cognitive

processing dysfunctions (Mazzocco, 2007; Gersten, Clarke, & Mazzocco, 2007).

Approximately 26% of students with learning disabilities exhibit problems in the area of

mathematics. Over 50% of students with disabilities have mathematics goals written into

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their individualized education programs (IEPs) (Kunsch, Jitendra, & Sood, 2007; Miller &

Hudson, 2007; Cass et al., 2003).

The term dyscalculia is a medically oriented term that describes a severe disability in

mathematics with medical connotations. When an adult suffers a brain injury and loses

abilities in arithmetic, medical professionals identify the loss of math skills related to the

neurological impairment as dyscalculia. An analogous term in reading is dyslexia, the loss

of reading skills that has medical and cognitive connotations.

Both mathematics difficulties and mathematics learning disabilities that emerge in

elementary school often continue through the secondary school years. Not only is a

mathematics disability a debilitating problem for individuals during school years, but it

also continues to impair them as adults in their daily lives (Maccini, Mulcahy, & Wilson,

2007; Adelman & Vogel, 2003; Cass et al., 2003). Almost one-half of the children who are

identified with severe mathematics difficulties in the fourth grade are still classified as

having serious mathematics difficulties three years later (Gersten & Jordan, 2005; National

Center for Learning Disabilities, 2006; Swanson, 2007).

It should be emphasized that not all students with learning disabilities or related disabilities

encounter difficulty with number concepts. In fact, some individuals with severe reading

disabilities do well in mathematics and exhibit a strong aptitude in quantitative thinking.

The identification and treatment of mathematics disabilities have received much less

attention than problems associated with reading disabilities (Fuchs, Fuchs, & Hollenbeck,

2007; Gersten, Clarke, & Mazzocco, 2007). For students with mathematics difficulties,

the mathematics curriculum in most general education classrooms does not pay sufficient

attention to learning differences in mathematics among students. Moreover, the general

education mathematics curriculum does not allot sufficient time for instruction, for guided

practice, or for practical applications. Further, mathematical concepts are introduced at too

rapid a rate for students who have difficulty with math. If students do not have sufficient

time to fully grasp a mathematical concept and to practice it before another mathematical

concept is introduced, they feel overwhelmed and become confused (Cawley & Foley,

2001; Butler et al., 2003). (Lerner & Johns, 2015, pp. 423-4)

Many changes have been made between the fifth and thirteenth editions of this section

of the textbook. Most notably is an explosion in neurological research into learning

disabilities that is indicated by the increase in these references cited from two in the

fifth edition to fourteen different references cited in the fifteenth edition. While much

of the medical discourse that is present in the earlier edition remains in the latest

edition, this time it is dominated by this “brain research” which wasn’t present in the

earlier edition. MLD is now defined as “a biologically based disorder and is related to

difficulties in cognitive processing and brain functioning.” This definition is supported

by the latest technology used in neurological research: “Research with functional

Magnetic Resonance Imaging (fMRI) shows these cognitive processing dysfunctions.”

Moreover, the use of prevalence rates, commonly used in medical discourse for illness

and disease, is notably present in the first three paragraphs of the section. Dyscalculia

is still mentioned in the latest edition but with an even more explicit medical

connotation: “The term dyscalculia is a medically oriented term that describes a severe

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disability in mathematics with medical connotations” (it is also, strangely, related to

brain injury in adults). And like the earlier edition, the word meanings, transitivity and

modality give the text an objective and impersonal feeling characteristic of medical

discourse. Note also the use of the phrase “identification and treatment of mathematics

disabilities” in the last paragraph, as if MLD were an illness to be cured.

Another one of the more notable changes between editions occurs in the title from

“Characteristics of Students with Mathematics Disabilities” to “Students With

Mathematics Difficulties and Students With Mathematics Learning Disabilities”. This

change is a reflection of the need by researchers to make a distinction between an

inherent (biologically based) mathematics disability versus one caused by

predominantly environmental factors (e.g. poor instruction or socioeconomic factors).

And the less strict category of mathematics difficulties enables researchers to study a

larger group of children who struggle with mathematics (Mazzocco, 2007). This

change seems to be an explicit response to clarify some of the inconsistent terminology

that is evident in the earlier edition and to acknowledge the challenge in identifying a

cause of students’ struggles with mathematics.

Another key change is the addition of the last paragraph. Here the authors suggest that

part of the cause of students’ difficulties in mathematics is institutional in nature: “the

mathematics curriculum in most general education classrooms does not pay sufficient

attention to learning differences in mathematics among students. Moreover, the

general education mathematics curriculum does not allot sufficient time for

instruction, for guided practice, or for practical application.” It is not teachers who do

not differentiate learning enough or do not give sufficient time for instruction, but the

curriculum. This is reflected by the grammar of the sentences in the way they

personify the curriculum as if it were the agent. It seems the curriculum is to blame for

some of the difficulties students have with mathematics. Similarly, the use of the

passive in the sentence “Mathematical concepts are introduced at too rapid a rate for

students who have difficulty with math” seems to shift some of the responsibility from

the teachers (by obscuring agency) to mathematics itself and the nature of the subject

as concept laden. Thus, the transitivity of the last paragraph seems to place some of the

responsibility for students’ struggles with math on the institution of schooling or the

discipline of mathematics. This suggestion of “environmental causes” is notably

absent from the first excerpt that squarely locates the source of students’ struggles

within the student. The last clause, “they feel overwhelmed and become confused,”

seems to position the struggling student as “victim” and mathematics as the

“perpetrator.”

CONCLUSION

Although medical discourse is still heavily present in the second excerpt (likely due to

recent research on the brain) and is rearticulated as the dominant discourse type, there

does seem to be an introduction of some creative discursive mixes. The change in the

title and the addition of the last paragraph introduce new discourses that shift some of

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the boundaries around the discursive practices on MLD—from a medical model that

locates the cause of the disability within the individual to one that notes other possible

causes located beyond the individual and within institutions. The expansion of the title

from “Mathematics Disabilities” to one that includes “Mathematics Difficulties” is a

clear acknowledgement that the source of mathematics struggles goes beyond the

individual and that environmental factors need to be considered. What was once

thought to be an inherent mathematics disability, may well have been caused by

something else (e.g. socioeconomic factors) and hence the need for the new category of

“mathematics difficulties” to account for this change in thinking. This analysis is thus

consistent with the lens of ableism which aims to expose the shifting boundaries often

used to box underachieving students (Borgioli, 2008) and reveal the fluid and

discursive nature of how a “disability” is defined.

References

Borgioli, G. M. (2008). A critical examination of learning disabilities in mathematics:

applying the lens of ableism. Journal of Thought, Spring/Summer, 131-147.

Fairclough, N. (1992). Discourse and social change. Cambridge: Polity Press.

Kincheloe, J.L., & McLaren, P. (1994). Rethinking critical theory and qualitative research. In

Norman K. Denzin & Yvonna S. Lincoln (Eds.), Handbook of qualitative research (pp.

433-488). Thousand Oaks, CA: Sage.

Lerner, J.W. (1989). Learning disabilities: Theories, diagnosis, and teaching strategies.

Boston: Houghton Mifflin.

Lerner, J.W., & Johns, B.H. (2015). Learning disabilities and related disabilities: Strategies

for success. Stamford, CT: Cengage Learning.

Mazzocco, M.M.M. (2007). Defining and differentiating mathematical learning disabilities

and difficulties. In D.B. Berch & M.M.M. Mazzocco (Eds.), Why is math so hard for some

children?: The nature and origins of mathematical learning difficulties and disabilities

(pp. 29-48). Baltimore: Brookes Publishing.

Rauscher, L., & McClintock, J. (1997). Ableism and curriculum design. In M. Adams, L.S.

Bell, & P. Griffen (Eds.), Teaching for diversity and social justice (pp. 198-231). New

York: Routledge.

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STUDENTS’ MODELLING PROCESS – A CASE STUDY

Minnie Liu


This article presents a case study in which three grade 8 students worked

collaboratively to solve a modelling task. Data indicate that students viewed the

problem situation from increasing levels of realistic perspective. At the beginning of

the modelling process, there was a disconnect between students’ mathematical

solution and the original situation and operated with numbers with a minimal

understanding of the problem situation. After multiple interventions from the

researcher, students were eventually able to draw on their lived-experiences and to

view the problem situation from an increasingly real world perspective and generated

a realistic mathematical solution.

INTRODUCTION

In a 1992 article, Dave Hewitt discusses the concept of “train spotting” – to solve

problems by looking for patterns and to predict solutions, to pay attention to the

numbers involved in the problem but move away from the original situation. Hewitt

suggests that what students seem to lack is not a fluency of their mathematical

knowledge but a connection between their mathematical solution and the original

situation, and the ability to recognize, access and apply this knowledge to solve

everyday problems. This results in problem solvers learning something about the

patterns, but not about the original mathematical situation. This disconnection between

their solutions and the original problem also contributes to the gap between reality and

the world of mathematics.

A widely cited example that has been used to demonstrate this disconnection between

students’ mathematical solution and the original situation is the bus problem

(Carpenter, Lindquist, Matthews, & Silver, 1983):

An army bus holds 36 soldiers, If 1128 soldiers are being bussed to their training site, how

many buses are needed?

Results of the study indicate that roughly 70% of students correctly carried out the

division that is required to produce a mathematical solution. However, more than half

of these students gave the answer of 31 buses or 31 buses remainder 12, and less than a

quarter of these students gave the answer of 32 buses. The results of this study, among

others (Greer, 1997; Verschaffel, De Corte, & Lasure, 1994), show that while students

often are capable of producing a mathematical solution, they tend to disregard their

mathematical solution from a real world perspective. These studies point out students’

inability to draw on their lived-experiences to verify the solution from a real world

perspective, and the gap between the real world and the world of mathematics.

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A possible way to address this gap between the real world and the world of

mathematics is through the use of modelling tasks. Modelling tasks are problems

situated in the real world. They require students to use mathematics as a tool to produce

a mathematical solution to the problem from a real world perspective. The process of

which students solve modelling problems can be described by modelling cycles (see

Figure 1). Students begin with understanding the situation. They then simplify the

situation and create a real model to represent the situation, mathematize the real model

in reality into a mathematical model in the world of mathematics, determine a

mathematical solution using the mathematical model, and verify the mathematical

solution by comparing it to the original situation (Borromeo Ferri, 2006; Mason,

1991). The processes of mathematization and interpretation connect reality with the

world of mathematics and are crucial steps in the modelling process. Research has

shown that mathematization is one of the steps in the modelling cycle where many

students experience difficulty, while interpretation and verification are often lacking in

students’ modelling processes (Galbraith and Stillman, 2006).

Figure 1: Modeling Cycle proposed by Borromeo Ferri (2006)

This study investigates the process of which students generate a realistic mathematical

solution to a modelling task and their ability to draw on their lived-experiences to

evaluate the created solution from a real world perspective. In this study, I intend to

take a closer look at the mathematization and interpretation steps in the modelling

process. Do students collect all information they need prior to building a real model

and mathematization? How do students decide what information is required in their

real model? Do they modify their real model during the modelling process? If so, how

do students decide what modifications are necessary? How do students verify their

mathematical solution? How do students decide whether to accept their mathematical

solution?

PARTICIPANTS AND METHODS

This study is part of a larger project which looks at students’ developments in

mathematical literacy skills through a modelling practice. In this article I present a case

study of three grade 8 (age 13-14) mathematics students’ modelling process. The three

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students are Hannah, Kevin, and Betsy1. At the time of this study, these students have

little experiences with such tasks.

This case study took place in a grade 8 mathematics class, where students were given a

modeling task and were asked to work as a group to solve the task, “The Door

Problem”. Students were asked to pay attention to their thoughts and approaches used

as they worked on the problem. Data include in-class observations, field notes,

impromptu interviews, post task interviews, and audio recordings of students’ work in

their group. The following is a summary of “The Door Problem”:

I have recently renovated my home office. As a final touch, I’m going to repaint and

decorate my office door. The plan is to paint and then decorate the door with Starbucks gift

cards. How much will this project cost me?

The mathematics required for students to complete the task is fairly minimal. Students

need to be able to do basic computations, determine the dimensions and surface area of

objects, etc. However, in order to be successful, students also need to draw on their

lived-experiences and make various decisions and assumptions about the situation. In

this case, these experiences include an understanding of the various tools one needs to

paint, the area of paint coverage per volume of paint used, etc. In what follows I present

these three students’ modelling process with a focus on mathematization,

interpretation and verification.

STUDENTS’ MODELLING PROCESS

Hannah, Kevin, and Betsy began by assuming the office door has the same dimensions

as the classroom door. Instead of measuring the door using a ruler or a meter stick, they

measured the door using gift cards, and planned on covering the door entirely with gift

cards. This was their first real model.

Upon determining the number of gift cards that fit on the door horizontally (12 cards)

and vertically (24 cards)2, the group quickly determined the number of gift cards that

can fit on one side of the door. At this point, students aimed to determine a solution to

the problem as quickly as possible. They did not have a very clear direction of where

they were going or what exactly they were trying to do. They multiplied 12 by 24 and

got 288, and used their estimation/rounding skills to determine the approximate

number of gift cards they needed to completely cover one side of the door. This was

their first attempt of mathematizing their real model into a mathematical one.

Prior to using the number of gift cards determined to calculate the cost for the project,

the group went back to discuss the way the gift cards were placed, and looked up the

possible cost for each gift card. Hannah first verified that Kevin measured the door by

1 Pseudonyms are used to protect students’ identities. 2 Kevin made a mistake with the measurements. The classroom door is 91.44cm wide and 203.2cm tall. A gift card is 8.57cm wide and 5.40cm tall. Kevin measured the width of the door using the short edge of the card, and the length of the door using the long edge of the card. Based on these measurements, Kevin should be able to fit more than 16 gift cards across the door.

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placing the cards vertically on the door, and soon enough, the group ran into the idea of

aesthetic. Since the gift cards have a horizontal design (see Figure 2), Betsy believed

that placing the cards vertically on the door would make the design look odd.

Figure 2: A Starbucks Gift Card

The discussion on how the cards should be placed lasted about ten minutes.

Meanwhile, students also did some re-measurements. Despite Betsy’s concerns, the

group decided to place the cards vertically and mathematized their idea and generated a

mathematical solution. This was their second attempt of mathematization.

Hannah So we, when put the cards on the door, we think, we are thinking of, we

R So

Hannah Vertical? Yeah. So we did that after that, and then, the bottom line we filled the door.

R Ok, so how many… how many cards?

Kevin Each card is five dollars, and we

Hannah Vertical. This is the price (points at the solution on a piece of paper), so we just put

R Wait a minute, 4803 times 5 is not 240.

Kevin 2040.

This is the group’s first mathematical solution. At this stage, students believed they

needed to fill both sides of the door with gift cards, ignored the cost aspects of the

problem, and failed to consider other tools required for this project. They did not verify

their solution either. They seemed to care more about finding a possible solution to the

problem than a realistic solution to the problem. In order to steer the group to a more

realistic solution than the one they have produced, the researcher intervened by

pointing out to the group that their current solution is extremely costly and they should

reconsider their approach to the problem. This is when Hannah and Betsy realized that

they were to decorate rather than to cover the door with gift cards, and that they should

consider the cost for painting the door as a part of the solution.

Hannah You don’t have to cover the whole door?

R Well I dunno, you decide.

3 The group re-measured the width of the door using the long edge of the card (10 cards), and multiplied 10 by 24 (the number of cards that could fit along the height of the door, measured using the long edge of the card). They then multiplied the answer (240) by 2 (2 sides of the door) to get 480 cards. While they have made a mistake here, this mistake does not affect their final solution to the problem.

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Hannah The paint?

Betsy Wait, it says, huh, paint the door, and then decorate the door.

Hannah Oooohhh…

As the group gained a better understanding of the problem situation, they went ahead to

discuss the possibility of various designs. After about a twenty-minute discussion, the

group settled on a particular design and determined the number of gift cards required,

the cost of the gift cards, and the cost of the paint required for the project. They then

used the information and generated a mathematical solution.

As the group talked to the researcher about their solution, they also wanted to use the

opportunity to verify their solution with the researcher. While the cost of the gift cards

and the cost of the paint seemed reasonable, the group has not yet considered painting

tools and adhesive as a part of the budget. Therefore, the researcher took the

opportunity and hinted to the group the needs of these tools.

R Okay, show me, show me the entire details. … show me what colour of paint I’m getting, do I need … br…

Hannah Oooo the brush!!!

R Ahh… the brush … And how am I getting the cards on (the door)?

Hannah Ooooooo

Rachel4 You need glue!

Hannah Oooooo

After another ten minutes of discussion, the group added painting tools and a roll of

tape to their budget and showed their work to the researcher again. At this point, the

students have created a detailed plan and included a can of green paint5, a paint brush,

28 gift cards, and a roll of tape in their budget. However, they were still missing a few

minor details in their plan. During the discussion, Betsy pointed out that they have not

yet considered sales tax. In order to cope with this, the group attempted to remove sales

tax from their equation by assuming they “live in a tax free country” or “Oregon,

USA6”.

Betsy Did you add tax?

Hannah (In a whiny tone) Why did you…

R Ooo did you add tax?

Hannah Okay… let’s say we live in a tax free country?

Kevin Oregon, USA.

4 Rachel is a student in the same class but in a different group. She overheard the conversation and answered the researcher’s question. 5 The can of paint has a volume of 946mL, which is sufficient to paint both the front and the back of the door twice. However, students’ solution does not explain how many coats they are painting the door. Therefore, it is difficult to judge whether they fully understood the concept of area of coverage. 6 Oregon, USA is a state in the USA that does not have sales tax.

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After some more discussion, Betsy accepted Kevin and Hannah’s idea of removing tax

from the real model in order to simplify the situation. In the end, the group further

modified their model by replacing the paint brush with a paint roller, and added a paint

tray and glue to their budget. They submitted what seems to be a reasonable budget

along with a design of the door.

Figure 3: Students’ Budget for the “Door Project” – Final Submission

STUDENTS’ MATHEMATIZATION PROCESS

This case study demonstrates novice task solvers’ modelling process to create a

realistic mathematical solution to a problem. In the beginning, students attempted to

create a mathematical solution based on their preliminary understanding of the

problem. They began by measuring the dimensions of the door and using these

measurements and the cost per gift card to generate a solution. Unfortunately, they

focused on a single aspect of the problem (cover the door with gift cards) and were

unable to draw on their lived-experiences to help them consider other aspects. This

hindered their process and resulted in an unrealistic solution (over $2000).

Furthermore, they did not verify their solution by comparing it to the original situation.

At this point, it can be said that there was a disconnect between students’ solution and

the original situation, and students approached the problem from a non-real world

perspective. They extracted numerical values from the original problem and operated

with these numbers to generate a solution, and ignored all realistic aspects of the

problem. Students were disconnected from the original situation. This is similar to

Hewitt’s (1992) “train spotting” concept.

After the researcher’s intervention, students went back to the original situation and

reconsidered the situation from a slightly more realistic perspective than before, and

created a design and included paint in their real model. After they mathematized their

real model and generated a mathematical solution, they verified their solution with the

researcher. In this sense, students completed the modelling cycle for a second time, and

were able to interpret the situation based on the context provided in the problem, which

is a slightly more realistic perspective than before. However, they were not yet able to

incorporate things that were not mentioned in the problem into their solution.

As the researcher attempted to push the group to further improve their solution,

students modified their real model by adding painting tools to their budget. They were

now able to expand their view of the situation to include things that were not

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mentioned in the original problem in their real model when prompted. The process of

modifying the real model, mathematizing, and determining a mathematical solution

happened very smoothly and naturally. This may be a result of the nature of the task.

Unlike other modelling tasks, the mathematical skills and calculations involved in

“The Door Problem” are very rudimentary. Students naturally went ahead and

determined the mathematical solution after building the mathematical model and

mathematizing the real model. Therefore, there seems to lack a clear distinction

between mathematical model and mathematical results in this case study.

As students discussed their solution with the researcher for the final time, one of the

students drew on her lived-experiences and pointed out the problem with tax.

Afterwards, these students further modified their real model and improved their

mathematical solution based on their knowledge about painting. This was the first time

the group was able to spontaneously draw on their lived-experiences and added other

realistic aspects to their solution without the researcher’s help.

To summarize, data in this case study demonstrate students’ multiple attempts to

generate a realistic mathematical solution to a problem based on their multiple levels of

realistic perspectives of the problem. In the beginning, students viewed the situation

from a rather non-realistic perspective and extracted numbers from the situation to

create an unrealistic mathematical solution. As the researcher intervened and as

students’ understanding of the situation deepened, students used the context given in

the problem to solve the problem from an increasingly realistic perspective. After

further interventions, students used some of their lived-experiences to improve their

solution. Eventually, students were able to spontaneously draw on their

lived-experiences in attempts to create a realistic solution, and were able to view the

problem from a real world perspective.

The mathematics required to produce a realistic mathematical solution to this problem

is rather rudimentary. However, students are required to bring in their

lived-experiences to make proper assumptions about the solution in order to be

successful in creating a realistic solution. In this case study, these three students relied

mostly on the researcher to help them draw on their experiences in order to produce a

realistic solution. When asked about this, students claimed that it is not that they do not

have these experiences, but they were rather overwhelmed by the situation and were

only able to focus on the mathematical procedures. They claimed that although they all

have some sort of painting experiences, they were not able to draw on these

experiences to help them solve the problem until they were reminded of these

experiences.

CONCLUSION

Novice task solvers go through the modelling process multiple times to generate a

realistic solution. This result is in line with Stender and Kaiser’s (in press) study. Data

in this study shows students’ ability to view the problem from an increasingly realistic

perspective as their understanding deepened and as they were reminded of the realistic

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aspects of the situation. Since this article presents results of a case study and includes

only a small sample of participants, further research is required before any

generalizations can be made. Further research areas include a deeper investigation into

the researcher’s role during the modelling process, students’ multiple levels of real

world perspective, and other possible reasons behind students’ inability to draw on

their lived-experiences during the modelling process.

References

Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the

modelling process. ZDM, 38(2), 86-95.

Carpenter, T., Lindquist, M., Matthews, W., & Silver, E. (1983). Results of the third NAEP

mathematics assessment: Secondary School. Mathematics Teacher., 652-659.

Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during

transitions in the modelling process. ZDM, 38(2), 143-162.

Greer, B. (1997). Modelling reality in mathematics classrooms: the case of word problems.

Learning and Instruction, 7(4), 293-307.

Hewitt, D. (1992, September). Train spotter's paradise. The Association of Teachers of

Mathematics, 6-8.

Mason, J., & Davis, J. (1991). Modelling with mathematics in primary and secondary

schools. Geelong: Deakin University.

Stender, P., & Kaiser, G. (in press). Scaffolding in complex modelling situations. ZDM.

Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical

modeling of school word problems. Learning and Instruction, 273-294.

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UNDERGRADUATE STUDENTS’ PERCEPTION OF

TRANSFORMATION OF SINUSOIDAL FUNCTIONS

Masomeh Jamshid Nejad


Trigonometry is one of the fundamental topics taught in high school and university

curricula, but it is considered as one of the most challenging subjects for teaching and

learning. In the current study, two different theoretical frameworks have been used to

examine undergraduate students’ perception of transformation of sinusoidal functions.

The results show that students did not grasp fully the concept of transformation of

sinusoidal functions.

INTRODUCTION

Trigonometry has a long history. Many ancient people used trigonometry for different

purposes. For example, Egyptians applied trigonometry to determine the correlation

between the lengths of shadow of a vertical stick with the time of day (see Figure 1).

Astronomers also used trigonometry to find longitude and latitude of stars, as well as

the size and distance of moon and sun. However, it was not until a Persian

Mathematician named Khwarizmi introduced trigonometric functions to the word.

Since then, trigonometry is one of the main topics in mathematics. Students often are

required to assign time for learning trigonometry, especially trigonometric functions,

because a strong foundation in trigonometric functions will likely strengthen their

learning of various mathematical topics, such as Fourier series and integration

techniques (Moor, 2010). It is shown that understanding calculus and analysis is

dependent on learning of trigonometric functions (Hirsh, Weinhold & Nicolas, 1991;

Demir, 2011). However, research studies (e.g. Weber (2005)) show that learning and

understanding trigonometric functions is a difficult and challenging task for students,

compared to other mathematics functions, such as polynomial functions, exponential

and logarithmic functions.

Figure 1: Egyptians’ use of trigonometry

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Kendal and Stacey (1997), for instance, assessed 178 high school students’

understanding of trigonometric functions through two different methods: ratio and unit

circle. The research shows that students had a better understanding of trigonometric

functions where functions were defined in terms of ratios of sides of right angled

triangles (students often taught SOHCAHTOA as a memory aid), rather than with the

unit circle method. These results are consistent with those of Palmer’s study (1980) in

which the students were randomly assigned either a ratio or unit circle instruction to

inspect which group grasped more trigonometric functions. The students in the ratio

classes outperformed their unit circle counterparts. Burch (1981) also concluded that

the participant students had difficulty interpreting trigonometric functions in the unite

circle, recognizing that x and y coordinates of a point on the unit circle are cosine and

sine values of corresponding angles compared with other determined trigonometric

functions in terms of right triangle. However, the results of Kendal and Stacey’s (1997)

study were in contrast with the finding of Weber’s (2005) study in which students

performed more successfully when they were learning trigonometric functions in the

unit circle than right triangle.

Brown (2005), Demir (2011) and Marchi (2012) suggest that students would better

understand trigonometric functions if they have more opportunities to take part in a

learning trajectory where both the unit circle and right triangles are utilized, rather than

learning trigonometric functions through instructional designs which focused on only

one of these two methods. To examine high school students’ understanding of

trigonometric functions, Brown (2005) developed a model of students’ understanding

of trigonometric functions within the geometry world of triangles and angles (in

degrees), and within the context of the unit circle. In this model, the sine and cosine of

an angle can be defined in three different ways: as ratios, as distances, and as

coordinates. The results of Brown’s (2005) investigations of 120 high school students

revealed that those students who were able to use and connect all interpretations (

ratios in right triangle, directed distances and coordinates) and moved flexibly between

them, could define trigonometric functions and were also better problem solvers.

However, Brown (2005) noticed that the majority of the students could still define

trigonometric functions and work effectively with only one view, instead of making

connections between all three representations. Brown concluded that most students

had an incomplete understanding of trigonometric functions.

In spite of all efforts has been done in the area of trigonometry, especially

trigonometric functions, still there has been some gaps in the literature review. There is

not research study focus on “how students transform sinusoidal function” and “what

students’ perception of transformation of sinusoidal functions is”. As such, the main

goal of this study is investigating answer for the above mentioned questions.

DATA COLLECTION AND ANALYSIS

The participants in the study were three undergraduate students, three females, from a

large North American university. The participants were selected from among students

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who had either completed calculus I course and were enrolled in calculus II (1

students) and in calculus I course at the time of the interview (1 students) in the

Mathematics Department. Another student finished calculus courses (Calculus I and

II) and she was enrolled in algebraic courses. The three participants were students who

had volunteered to participate in the study after I had made a general request to the

whole class.

A 60-minutes task-based interview was conducted and students were required to

complete two types of interview tasks:

1) Recognizing sinusoidal functions (the tasks were: Task 1: , Task 2:

and Task 3: .

2) Assigning coordinates (Task 4: and Task 5: ). For

the first type of tasks the sketches indicating the sinusoidal graphs were given and

students were required to identify the transformed sinusoidal functions represent the

given graphs (see Figure 2). For the second types of tasks, a particular transformed

sinusoidal functions were given along with a wavy displace (see Figure 3) and students

needed to assign coordinate on the wavy curve in a way that describe the given

function.

Figure 2: Snapshot of task 2

Figure 3: a wavy displace for the second type of tasks

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The collected data in this study were analyzed and interpreted according to two

theoretical frameworks. Mason’s theory of shifts of attention is first framework to be

used. Mason’s theory provides opportunity for the researcher to study the critical role

of attention and awareness in learning and understanding mathematics and in particular

the concept of transformation of sinusoidal functions. Mason divided attention

structure in to 5 categories: 1) holding wholes; 2)discerning details, 3)recognizing

relationships; 4)perceiving properties; and 5)reasoning on the basis of agreed

properties. Mason’s theory provides opportunities to investigate whether students

focus their attention, for instance, on detail of the transformed sinusoidal

functions/graph, or they attempted to find relationship between symbolic and graphical

representation. The second theory is Presmeg’s (1989) visualization theory. She

identified five different category of visual imagery: 1) concrete, pictorial imagery; (2)

pattern imagery; (3) memory images of formulae; (4) kinesthetic imagery; and (5)

dynamic (moving) imagery. As a broad theory, it is used for investigating students’

visual mental constructs since the participants applied their imaginary skills in

different occasions when they completed the interview tasks. Visualization will allow

investigating mental process in the mind of participants which can be identified

through drawing on the paper, or manipulating a diagram on the computer screen.

Reviewing students’ answers and reading the transcripts lead the researcher into two

main themes: “recognizing the period (B)” and “recognizing the phase shift (C)” (.

Please note that in all the 5 interview tasks the participants were required to connect the

period of the given sinusoidal function or the sine curves to a coefficient of x in a sine

or a cosine function. For brevity, we refer to this connection as “recognizing the

period” (see Figure 4). In the Tasks 3 and 5 the participants were required to recognize

the phase shift/horizontal shift. From the canonical function ( the phase shift is

obtained by determining the change being made to the x-value (see Figure 5). The

displacement will be to the left if the phase shift is negative and to the right if the phase

shift is positive. For brevity, we refer to this connection as “recognizing the phase

shift”.

Figure 4: Recognizing the period

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Figure 5: Recognizing phase shift/horizontal shift

Recognizing period

When students completed mathematics Task 1, they all initially were confused and

they all agreed that the function described the given graph (see Figure 2) was

For instance, Rose expressed:

“…It is sine graph, because it starts at 0. it should be . The sine graph

starts at 0 and then π and 2π but this one is o, π, 2π /3. This is half of sin graph... The period

here is π while it is 2π in the original sine curve.”

Emma also stated:

“….because a normal sine graph is between 0 and 2π but in this one, the graph is

compressed by half so it means that it is going to be ”

As it is clear from the above excerpts, the students focused their attention on the details

embedded in the given graph and they tried to fin relationship between the graphical

representations with the symbolic representation of the transformed function. They

were using pictorial imagery of canonical sinusoidal functions (“…because a normal

sine graph is between 0 and 2π”). Sally also used her Kinaesthetic imagery (see Figure

6) while she attempted to described the function, although similar to the other students

she was unsuccessful.

Figure 6: Sally’s kinaesthetic visualization

After observing the graph of students noticed their mistake and they

all found a correct transformed function for the given graph. Since then, for the rest of

interview tasks (such as Task 4) when the coefficient of x in the transformed sinusoidal

function was a whole number, students successfully completed the tasks. However, in

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other tasks such as Task 2 and 3, only one student was able to recognize the period

successfully. Emma stated:

“…the graph should go over 2π… it is f(x)= sin(2/3x) because the period is 2π/B and the

period ends here…here is 3π”

As it is clear from the statement, Emma tried to discern details from the given graph to

identify the transformed function. She also used her memory image of formula (2π/B)

in order to find period. Besides, identifying period, students needed to recognize phase

shift as well.

Recognizing phase shift

Tasks 3 and 5 involved phase shift. However, none of the participants successfully

realized phase/horizontal shift. As an example, for task 5 (f(𝑥)=𝑐𝑜𝑠(3𝑥 − 𝜋 /4)) Sally

expressed:

“ … because the amplitude is not changing so here is 1 and -1… original period is 2π…we

have transformation on x to the right direction. So we move the graph over by π /4…”

While she focused on the properties of canonical sine function (the amplitude is not

changing so here is 1 and -1… original period is 2π), she was unable to recognize π /12

as a proper horizontal shift. In other words, if one thinks about the standard formula of

transformation of sinusoidal function: 𝑓(𝑥)=𝑠𝑖𝑛/𝑐𝑜𝑠(𝐵(𝑥±𝐶)+𝐷) and

𝑓(𝑥)=𝑠𝑖𝑛/𝑐𝑜𝑠(𝐵𝑥±𝐵𝐶)+𝐷, while BC correspond with phase shift, she recognize it as

“C.” It was not only Sally who found phase shift incorrectly; the other two participants

were unsuccessful as well.

RESULTS

Result of the study show that the three students participated in this research study

focused their attention mostly on the: discerning details from the given transformed

sinusoidal function or graph, finding relationships between the graphical

representation and symbolic representation of the transformed functions, and

perceiving properties of sinusoidal functions while transformed sinusoidal functions.

They also relied on concert imagery of the canonical sinusoidal functions, pattern

imagery of the transformed functions and kinesthetic imagery. However, data show

that students were unsuccessful in realizing period when the coefficient of x was not a

whole number in the transformed sinusoidal function, although they were able to

identify period when the coefficient of x was a whole number. The data reveled that

none of the participants successfully realize proper phase shift from the given

transformed function or their corresponding graphs.

References

Brown, S. A. (2005). The trigonometric connection: Students' understanding of sine and

cosine. (Unpublished Ph.D. Dissertation), Illinois State University: USA.

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Demir, O, (2011). Students’ concept development and understanding of sine and cosine

functions. (Unpublished doctoral dissertation), University of Van Amsterdam.

Kendal, M., & Stacey, K. (1997). Teaching trigonometry. Vinculum, 34(1), 4-8.

Weber, K. (2005), Students’ understanding of trigonometric functions. Mathematics

Education Research Journal, 17(3), 91-112.

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COHERENT AND DIVERGING DISCOURSE IN

MATHEMATICAL ACTIVITIES WITH DYNAMIC GEOMETRY

Oi-Lam Ng


This paper discusses the detection of coherent and diverging discourse in

mathematical exploratory activities using dynamic geometry environments (DGEs).

The data is drawn from a larger study examining patterns of communication within

pair-work mathematical activities using touchscreen DGEs. A

thinking-as-communicating approach is used to analyse communication involving

pairs of high school calculus students’ exploration of the area-accumulating function

using touchscreen DGEs. Results show that the students integrated speech, gestures

and touchscreen-dragging synchronically and diachronically to engage in coherent

and diverging discourses. In particular, it is shown that conflicting ideas can be

communicated non-linguistically. This paper raises questions about new forms of

communication mobilised by touchscreen DGEs.

INTRODUCTION

During the past decade, the evolution of digital technology has resulted in an integrated

system for communication and representational expressivity, opening possibilities for

social mediation of ideas (Hegedus & Moreno-Armella, 2014). In particular, the

introduction of digital technologies in schools has given rise to new ways of doing and

representing mathematics. Recently, a number of research has focused upon the haptic

and dynamic nature of touchscreen digital technologies and how this affords new

modes of thinking and learning (see for example, Falcade, Laborde & Mariotti, 2007;

Sinclair & de Freitas, in press). Related to this, my previous work drew on the effect of

dynamic geometry environments (DGEs), particularly when presented on touchscreen

devices, for facilitating a multimodal communication about calculus. One of my

significant findings is that touchscreen-dragging on DGEs enabled students to

communicate temporal relationships with their fingers, where verbal communication

was also transformed in the presence of dragging (Ng, 2014). As touchscreen

technology continues to enhance digital experience of learners, its affordances for

mathematical learning, especially as it is complemented with DGE affordances, is an

important area of study. There is a need to understand what mathematics is

communicated and how it is communicated in an era of touchscreen DGEs. This study

focuses the latter, by investigating high school students’ discourse in mathematical

activities with DGEs. More specifically, this study addresses how coherent and

diverging discourses are communicated in mathematical exploratory activities with

DGEs.

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The study adopts a learning as participation perspective that considers learning

mathematics as a social activity (Lave & Wenger, 1991). This perspective suggests that

learning is located neither in the heads nor outside of the individual, but in the

relationship between a person and a social world. Sfard’s communicational framework

(2008) is based upon the learning as participation perspective and highlights the

communicative aspects of thinking and learning. Sfard (2008) uses the term

commognition to encompass thinking (individual cognition) and interpersonal

communication as manifestations of the same phenomenon.

Sfard theorises mathematical thinking as the process of individualising or developing

one’s mathematical discourse. She proposes four features of the mathematical

discourse, word use, visual mediator, routines, and narratives, which could be used to

analyse mathematical thinking and changes in thinking. For the purpose of this paper,

the first three features will be used for analysing the use of language, gestures, and

dragging in one’s the mathematical discourse. Word use is a main feature in

mathematical discourse; it is “an-all important matter because […] it is what the user is

able to say about (and thus to see in) the world” (p. 133). As a student engages in a

mathematical problem, her mathematical discourse is not limited to the vocabulary she

uses. For example, her hand-drawn diagram and gestures can be taken as a form of

visual mediator to complement word use. Routines are meta-rules defining a discursive

pattern that repeats itself in certain types of situations. In learning situations, teachers

may use certain words or gestures repeatedly model a discursive pattern, such as

looking for patterns and what it means to be “the same”. Hence, gestures can be taken

as both a routine for defining a discursive pattern that repeats itself, and a visual

mediator in a student’s discourse. A student may use gestures to reason the change of

tangent slope, in which it becomes a routine, or to mediate the variance of the tangent

slope, in which it becomes a visual mediator. This communication can be interpersonal

when it is directed to another student or intrapersonal when it is directed to oneself.

According to Sfard (2009),“Using gestures to make interlocutors’ realizing procedures

public is an effective way to help all the participants to interpret mathematical

signifiers in the same way and thus to play with the same objects” (p.198).

Sfard conceptualises learning mathematics as a change in one’s mathematical

discourse. This can be observed through one’s word use, visual mediator, routines, and

narratives. Diverging discourse arises as inconsistent words, visual mediators, routines

or narratives are used in communication. Communicating in diverging discourses may

lead to commognitive conflicts—the encounter between interlocutors who use the same

mathematical signifier in different ways or perform the same mathematical tasks

according to different rules.

METHODOLOGY

The study was part of a larger research project aimed at studying high school calculus

students’ patterns of communication during pair-work mathematical activities using

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touchscreen DGEs. The participants are five pairs of calculus students enrolled at a

culturally diverse high school in Western Canada. In this paper, I focus on the data

concerning an exploratory activity about the area-accumulating functions. The purpose

of the present study is to investigate how coherent and diverging discourses are

communicated in mathematical exploratory activities with touchscreen DGEs. To

answer this research question, the participants needed to have some prior experience

working in an exploratory setting with touchscreen DGEs. At the time of study, the

participants had just completed the differential calculus component of the course,

where key concepts in differential calculus had been taught with consistent use of a

class set of touchscreen DGEs in their regular calculus classroom. During these

lessons, students were guided by the classroom teacher to explore and discuss calculus

concepts as presented on the DGEs in pairs before the traditional lectures are given.

The task took place outside of school hours in the participants’ regular classroom. It

invited the five pairs of participants, who were regular partners in their class, to discuss

a pre-designed sketch presented on the touchscreen DGE application, SketchExplorer

(Jackiw, 2011). The sketch contained five pages and was designed to facilitate

exploration of the area-accumulating function, A(x)=a∫xf(t)dt. As the participants had

yet to learn this in their regular classroom, the goal of the sketch was to introduce the

idea of “area as a function”, and this can be achieved when the students were able to

relate the set of “green traces” as the graph of area under ‘f’ from ‘a’ to ‘x’ (Figure 1).

The researcher turned on the videotaping function of the camera facing the

student-pairs and left the room, until the students finished exploring and discussing.

Each student-pair took between 30 to 45 minutes for completing the task.

(a)

(b)

Figure 1(a) A page representing the area-accumulated under f(x)=1, a=0 and (b) under

f(x)=sinx, a=0. As ‘x’ is dragged, area A(x)=a∫xf(t)dt is traced dynamically in green.

As it was necessary for this study to capture student pairs’ utterances, gestures, and

dragging actions while interacting with DGEs, special methodological decisions are

made around the transcribing of data and its organisation. Unlike conventional

transcripts which informs only “who said what”, I included a record of “who gestured

what” and “who dragged what” in the transcript. As a transcript convention, certain

parts of the transcript were underlined and double-underlined to keep track of the

words spoken while a gesture or dragging action was performed simultaneously by one

of the students. In addition, two kinds of analyses, synchronic and diachronic, were

carried out in order to address the coherency of discourse in the student pairs’

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communication. According to Arzarello (2006), these two kinds of analyses are needed

to fully understand a semiotic activity. Synchronic analysis enables the study of

relationship among different semiotic sets activated simultaneously. Diachronic

analysis studies the same phenomenon in successive moments. To this end, a

synchronic lens is used to analyse the coherency of the participants’ discourse at one

moment of calculus communication, and a diachronic lens is used to observe how

participants respond to each other successively using speech, gestures and dragging,

and whether diverging discourses have arisen over time.

RESULTS

Data from two student pairs were selected in this paper to exemplify the multimodal

communication used generally by all student pairs, and in particular, to illustrate

characteristics of coherent or diverging discourse during the activity.

From coherent to diverging discourses

The transcript below was taken between Larry and Ivy’s discussion while they were on

the page with f(x)=1. At Turn 15, the students were 2 minutes into their discussion.

15 Larry: So, ‘A’ is probably the dotted line, right? Oh, so like it's like from here, to here. <Larry drags ‘x’>

16 Ivy: Oh. Oh, so that's the area. <Larry drags ‘x’>

17 Ivy: <Ivy drags ‘x’> <Ivy gestures> 'Cuz... one and one...one times one is one... <Ivy

drags ‘x’> one times two is two...one times three is three...so that's... I guess that slope is the area of that. Shaded. <Ivy taps Larry’s arm> Try the next one.

18 Larry: Oh. Oh, so that's the area. <Larry drags ‘x’>

At Turn 15, Larry suggested that ‘A’ was “probably the dotted line,” while dragging

‘x’ around. Then, while Larry was still dragging, Ivy responded verbally, “Oh. Oh, so

that’s area” (Figure 2a). Her use of pronoun “that” seemed to be referring to Larry’s

“dotted line” in his previous utterance, which would mean that Ivy was communicating

the dotted line as area. At Turn 17, Ivy took on the role of dragging and used a series of

gestures while speaking of “area” numerically. She talked of three different moments

of calculating area under f(x)=1 from a=0 to ‘x’ as she dragged ‘x’. She first dragged

‘x’ to x=1. Then, while she said “one times one is one,” she used her middle finger to

point towards the draggable point ‘x’, which was at (1,0) at the time, followed by

pointing towards the point (1,1) (Figure 2b). Next, she dragged ‘x’ continuously from

x=1 to x=3 and uttered, “one time two is two, one times three is three” (Figure 2c). It

was clear that Ivy was stating how the area of the rectangles could be calculated

numerically. The timing of her utterance corresponded to the very state of the rectangle

as ‘x’ was dragged continuously, showing that she was coordinating her dragging and

speech. She added the word “shaded” at the end of, “I guess that slope is the area of

that,” to clarify that she was talking about the shaded area. Ivy ended her utterance with

“Try the next one. ‘Kay”. The significance of this utterance was that Ivy asked Larry to

turn the page, and Larry agreed to do so. Not only did he turn the page at Ivy’s request,

Larry uttered the exact same words “so that’s area” (Turn 18) in response to Ivy’s

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explanation. All this suggest that the two students were “on the same page” both

literally and figuratively—in that they were both ready to move onto the next page.

(a) Ivy’ gesture accompanied by the utterance, “Oh. Oh, so that’s the area.”

(b) Ivy points at the point (1,0) and then (1,1) while uttering, “one times one is one”.

(c) Ivy drags ‘x’ from x=1 to x=2 and then x=3, all the while speaking of the calculation of area.

Fig. 2 (a-c) Selected snapshots of Larry and Ivy’s gestures and dragging (Turns 15-18)

A minute later, while they were on Page 2, which showed the function f(x)=x, the

students noticed that the sign of ‘x’ and ‘y’ affected the sign of the “area”. Turns 23 to

24 briefly illustrates the students’ discourse about the sign of the “area”.

23 Ivy: Mmm... So yeah, the area becomes negative because that's a negative ‘x’, right? <Larry drags ‘x’> <Ivy gestures>

24 Larry: Mmm… So when it's at zero, it'd be zero. <Larry drags ‘x’>

The students talked about “area”, in particular, the sign of “area” in the above

transcript. At Turn 23, Ivy’s speech accompanied Larry’s prolonged dragging of ‘x’. In

this utterance, Ivy used the conjunctive “because” to suggest a causal relationship

between the sign of the area and the sign of ‘x’. Perhaps, she had noticed that when the

sign of ‘x’ was negative, the area under the function f(x)=x from a=0 to ‘x’ was

negative. Turn 24 showed Larry’s response to Ivy. He first stopped dragging and said,

“Mmm.” Then, he resumed dragging of ‘x’, dragged it towards x=0, and said, “So

when it’s at zero, it’d be zero.” It was suggested that Larry was naming the two

mathematical objects that Ivy had talked about previously with the pronoun, “it”. This

would mean that he was communicating the idea that, “when [‘x’] is at zero, [the area]

would be zero”. Thus, his discourse integrating speech and dragging of ‘x’ towards

zero was coherent with Ivy’s discourse.

Recall that Larry and Ivy were in agreement in the beginning about when to turn to the

next page. In contrast, at Turn 39, the two students showed disagreement when Larry

wanted to turn from Page 3 (with f(x)=x2) to Page 4 (with f(x)=sinx):

39 Larry: I guess. K, and then there's sine. <Larry drags ‘x’> <Ivy gestures>

40 Ivy: Wait, wait. <Larry drags ‘x’>

As shown in Turn 39, Larry was ready to turn the page when he said “K, and then

there’s sine,” as he placed his finger on the page tab “Area under sine.” Apparently, Ivy

was not ready to move on, as she said, “wait, wait”. This meant that she was hoping to

stay on the page and perhaps to find out something that Larry did not. Moments before

this, Larry and Ivy had agreed that the green traces on this page were of “third degree”.

Perhaps Larry was satisfied with this discovery, but Ivy was not. In fact, it was

observed that their thinking began to take different turns from this point forward. Larry

and Ivy began to develop diverging discourses after Ivy’s “wait, wait”. More

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specifically, Ivy seemed interested in finding invariance across all pages, but Larry did

not seem interested in pursuing it. For example, towards the end of the activity, Ivy

noticed something “strange”, while Larry seemed satisfied with their discussion. The

students left their discussion on that note at the end of the mathematical activity.

Commognitive conflicts communicated through dragging

The previous shows the change of Larry and Ivy’s discourse from a coherent to a

diverging one at the end of their exploration. The analysis to follow capture the use of

touchscreen-dragging as a mode of communication, in particular, that gave rise to

commognitive between Huang and George.

At the 5-minute mark of their exploration, Huang and George exploited the

multi-touch function of the DGE to drag ‘a’ and ‘x’ simultaneously and respectively.

After 5 seconds of dragging simultaneously and silently by the two students, Huang

broke silence with his question, “What are you doing George?” Since both students

had already been dragging for 5 seconds, it seemed unlikely that Huang had just

realised that George was also dragging alongside him, and so this analysis did not

regard his question, “what are you doing”, in a literal sense. The situated meaning of

this question seems to be that Huang was in disagreement with what George was doing.

There might have been a change in his thinking from 5 seconds ago, since he had not

raised this question back then. It also suggests that there existed a conflict in the

students’ discourse, even though they were not actually speaking. In Sfard’s terms, this

commognitive conflict, as reflected in the students’ dragging routines, led to Huang’s

question about what George was doing.

(a) Huang and George each dragged a point on the iPad towards each other

(b) Huang and George each dragged a point on the iPad away from each other

Figure 2 (a,b). Huang and George dragged ‘a’ and ‘x’ respectively for 5 seconds before

Huang broke silence, “what are you doing, George?”

Upon Huang’s question, George refrained himself from dragging and, in so doing, let

Huang be the sole dragger on the iPad. George’s response could be taken as his attempt

to resolve the commognitive conflict in the students’ dragging routines, by voluntarily

refraining himself from actively participating in the mathematical activity. Hence,

George watched still as Huang dragged ‘x’ for another 9 seconds, twice changing his

direction of dragging. Importantly, the present analysis shows that commognitive

conflicts could be present in non-verbal communications. The students might have

communicated diverging discourse through their dragging routines as indicated by

Huang’s remark. The analysis shows that George’s means to resolve the conflict was

also non-verbal. Therefore, it can be said that the students had relied on non-verbal

communications for developing their mathematical discourses during this episode.

MEDS-C 2015 Ng

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DISCUSSION

This paper raises important questions about the way coherent or diverging discourses

are communicated in pair-work and group-work mathematical exploratory activities

using DGEs. The analysis shows how different attention towards the DGE can serve to

diverge two students’ discourse. The student pairs were told to discuss what they saw

in the sketch containing five page tabs: the different pages designed in the sketch

contributed to the student pairs’ agreement and disagreement as they had to decide

when to move on to a new page or go back to a previous one. For example, in Larry and

Ivy’s early exploration, Ivy tapped Larry’s arm to signal Larry to turn the page, which

was agreed by Larry. In contrast, moments after they turned to Page 2, Larry and Ivy

showed disagreement when Larry tried to turn to the Page 3 and Ivy’s called out,

“wait”. Using a communicational framework, agreeing on when to turn a page can

reveal what the student/pair was attending to, especially by observing the

student/pair’s discourse after a page was turned. Therefore, these moments can provide

useful pointers as to what a student/pair is thinking mathematically. In Larry and Ivy’s

case, Ivy was likely interested in finding more about what she later called “strange”

about the area-accumulating function, yet Larry was not interested in pursuing it but

more interested to move on to the next page. Indeed, as shown in a later analysis, Larry

seemed quite satisfied with being able to state the sign of A(x) and was not interested in

exploring the shape of A(x). This suggests the students’ discourses have diverged at

the end of their exploration.

Besides the action of turning a page, the student pairs also used particular words to

communicate in coherent and diverging mathematical discourse. There were ample

evidence showing that the student pairs complemented each other’s communication

such as finishing their partners’ utterances verbally. Besides word use, their gestures

and dragging complemented their partners’ also. The simultaneous and

complementary use of dragging and speech by two different persons indicates a

coherent way of mathematical thinking by both discussants. Hence, the student pairs

were engaging in one coherent discourse as opposed to two divergent ones. In contrast,

at one point, Huang and George were engaging in conflicting discourse while both

students were dragging simultaneously. After dragging ‘a’ and ‘x’ respectively for five

seconds simultaneously, Huang asked George, “what are you doing,” implying that he

did not understand what George was doing. This means that a commognitive conflict

may have arisen.

In summary, I combine Sfard’s commognitive approach to enrich the understanding of

how student-pairs engage in coherent and diverging discourse within mathematical

exploratory activities with touchscreen DGEs. In particular, I identified the kinds of

words, gestures and touchscreen-dragging that were characteristics of the coherent and

diverging discourses. Consistent with the findings of the larger research that this study

was situated, a multimodal view of communication was essential for framing the

results of the study. Examining student pairs’ speech, gestures and dragging

synchronically provides information on whether or not they are communicating

MEDS-C 2015 Ng

78

coherently at the given moment, while a diachronic analysis helps to see whether or not

their discourse has diverged during the course of the activity. Diverging discourse may

be detected by a disagreement of when to turn the page, questioning the action of one’s

partner, non-complementary use of speech, gestures, and dragging, as well as the use

of pronouns in singular form (“I” or “you”) as opposed to a plural form (“we”). This

developed framework for understanding mathematical discourses within exploratory

activities with DGEs is especially relevant for classroom teaching because it informs

when a pair or group is making progress in their development of the mathematical

discourse collectively or individually. In particular, this framework can be used by

classroom teachers to decide when intervention is needed to help bring the discourse

back in sync, which is what Wells (2014) terms, “teaching from the sideline”. Future

research should consider adopting a multimodal communicational framework to

examine the coherency of discourse in other mathematical activities, and those situated

in other tool-based learning environments to address the role of tools in thinking and

learning.

References

Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de

Investigación en Matemática Educativa, 9, 267–299.

Falcade, R., Laborde, C., & Mariotti, M. (2007). Approaching functions: Cabri tools as

instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317-333.

Hegedus, S., & Moreno-Armella, L. (2014). Information and Communication Technology

(ICT) Affordances in Mathematics Education. In S. Lerman (Ed.), Encyclopedia of

Mathematics Education SE - 78 (pp. 295–299). Springer Netherlands.

Jackiw, N. (2011). SketchExplorer [computer software]. Emeryville, CA: Key Curriculum

Press.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation.

Cambridge, UK: Cambridge University Press.

Ng, O. (2014). The interplay between language, gestures, dragging and diagrams in bilingual

learners’ mathematical communication. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan

(Eds.). Proceedings of the Joint Meeting of PME 38 and PMENA 36 (Vol. 4, pp.

289-296). Vancouver, Canada: PME.

Sinclair, N., & de Freitas, E. (in press). The haptic nature of gesture: Rethinking gesture with

new multitouch digital technologies. Gestures.

Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses,

and mathematizing. Cambridge, England: Cambridge University Press.

Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in

Mathematics, 70(2), 191–200

Wells, K. (2014). A conversation-gesture approach to recognizing mathematical

understanding in group problem solving (teaching from the sidelines). (Unpublished

doctoral dissertation). Simon Fraser University, Canada.

MEDS-C 2015 Noble

79

IDENTITY AND MATHEMATIS: A PRELIMINARY

EXPLORATION

Tanya Noble


This preliminary exploration considers the Affinity-identity students have with

mathematics. The relevance of this became apparent as I noticed my own

Affinity-Identity reaffirm my growth into the role of a mathematics education

researcher. I explore possible ways to identify affinity and develop a picture of how

identity of the learner and the subject to be learned can be identified. Development of

identity is a communicative practice situated in cultural experiences and therefore an

analysis of both praxis and individual communication could offer deep insight into

influences on student identity as they seek affinity with the identity projected onto

mathematics.

IDENTITY AND MATHEMATICS

At the intersection of mathematics and student7 learning of mathematics there exists a

boundary where mathematical knowledge is transferred through communication

(verbal, physical, written and symbolic). I suggest it critical to study the tangible yet

elusive gap between student identity and the identity assigned to mathematics. The

praxis (Grootenboer, 2013) of the mathematics classroom and its influence on the

formation of student identity with respect to mathematics must be the starting point of

analysis.

As I deepen my own insights of the important contribution identity brings to learning I

developed a curiosity about the identity students, as a collective, have come to apply to

mathematics. It is this identity that students seek affinity with in order to develop deep

connections with the subject. Does mathematics have an identity with which students

build affinity? What tools are available for the researcher to recognize the identity

students have granted mathematics?

This is a preliminary exploration of student identity to inform the relevance and

importance Affinity-identity brings to the relationship between students and the

mathematics. This exploration seeks to reflect on the significant insight different

Identity lenses offer to the development of a strong identity framework to explore the

importance of Affinity-Identity.

7 Throughout this paper I was tempted to use the term learner in place of student. I

perceive learners to have a more active role in their studies and that this is a better term

for a socio cultural perspective. The term student builds an image of an individual

receiving information. I kept with student only because all of the research on identity

was “student identity” rather than “learner identity”.

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LEARNING AS COMMUNICATIVE: WEAVING MY EXPERIENCES AS

LEARNER AND TEACHER

The nature of learning is communicative (Sfard & Prusak, 2005) and requires a

socio-mathematical perspective linked to the social practices people engage in to

produce or construct mathematical knowledge and facts (Restivo, 1997, p. 16).

Grootenboer (2013) discusses praxis as a holistic view of the culture of the

mathematics classroom where the students’ broader mathematical behaviour is

engaged. If a student does not have affinity with mathematics they may perceive an

insurmountable gap to successful learning of mathematics.

The study of student identity has coincided with my own transformation from

practicing teacher to theoretical observer. Through personal reflection I recognized

similar patterns inherent in my own learning experience and in the experience of

students in a variety of educational institutions. Through my own self-study I reflected

on significant moments in my journey where I witnessed affect coincide with a

reification of my own identity as it intertwined with the identity I perceived as that of

the mathematics education community.

A high school mathematics student stepping into a mathematics classroom or a high

school mathematics teacher attempting to make a move into the theoretical world of

mathematics research, both are students who seek affiliation or closeness with a group

to share experiences, ideas, and challenges that enhance their learning (Cobb, Gresalfi,

& Hodge, 2009). Mathematics curriculum is steeped in a strong history of strict criteria

that dominates the identities of scholarly mathematics. This curriculum arguably

isolates students who do not follow the clear traditional paths to the study of maths.

D’Ambrosio (1987) prompts consideration that time has seen growth of mathematics

into two clearly distinct branches: 1) “Scholarly” mathematics, which was

incorporated in the ideal education of Greeks; and 2) “practical” mathematics, reserved

for manual workers (p.15). These branches are evidenced in both the mathematics

curriculum and the mathematics education research community.

The tension existing between student identity and the perceived identity of

mathematics is foundational to student’s beliefs about learning about their abilities in

the mathematics classroom. The identity of the student is formed within an interpretive

system seeded with re-contextualization of the students past experiences with

mathematics. Nixon (1994) observed that education mediates between the culture of

the individual student and the classroom culture. In the mathematics classroom,

teachers act as agents of mathematical identity; in this role they make choices that

communicate identity of a successful mathematics student. These mathematical traits

are reinforced through assessment practices as well as recommendations about student

ability and trajectory (Nixon, 1994; Grootenboer, 2013). Through praxis the invisible

aspects of the cultural core are conveyed to the student. In order for students to have an

affinity with mathematics they must find commonalities between aspect of their own

MEDS-C 2015 Noble

81

core identity (Sfard & Prusak, 2005) and the identities granted power through scholarly

mathematics.

Experience in the mathematics classroom profoundly influences how students identify

with mathematics. Identity is constructed through connections to mathematics through

praxis. The notion of the external influences on identity appears in works by Sfard and

Prusak (2005), and Cobb et al. (2009). Burton (1994) writes that if mathematics were

to be experienced as a searching, hesitant, intuitive area of study, open to interpretation

and challenge, these would be a much greater identification with its style and ideas by

pupils, both males and females, different classes and different races (p.79). In this

sense mathematics should be presented in different ways within different contexts.

Nixon (1994) views the changes in mathematics education from three broad

perspectives: (a) those related to the nature of mathematics as a discipline; (b) those

concerned with research about teachers and the teaching of mathematics, and (c) those

concerning pupil perspectives (p. 7). The dominant identity in the mathematics

classroom has traditionally been rooted in the Institutional Identities and Discourse

Identities of scholarly mathematics (D’Ambrosio, 1997; Gee, 2001). My own

reflections through personal experiences have guided my interest in the emphasis of

pupil perspectives fostered by the current Reform movement in education. The reform

movement advocates for an expansion of a set of shared experiences among students to

build interactions in the gap that exists between students’ mathematics identity and the

identity they perceive is that of mathematics (Nasir, 2002, p. 238). Experiences

fostered through the reform movement are where viable possibilities provide for a

restoration of balance among the various identity lenses. The reform movement

follows classroom cultures that nurture social interactions necessary to access Gee’s

A-Identity (Affinity Identity).

PRELIMINARY DATA COLLECTION

Data was collected through a questionnaire distributed on the first day of school in

September to 90 grade nine students. Students were asked to reflect on two questions:

What is mathematics? and What does it mean to learn mathematics? The student

responses implied there was a range of mathematical experience in the sample:

“Since you’re looking at many different ways for a solution to a problem, it helps you look

in different perspectives to other questions in life” (Grade 9 student, Vancouver, BC)

“Learning basic skill to succeed in life, but at this point everything we learned in high

school has nothing to do the real world” (Grade 9 student, Vancouver, BC)

“To learn mathematics means to understand it. You can’t learn something that you don’t

understand” (Grade 9 student, Vancouver, BC)

What is this “it” that students refer to?

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WHERE TO GO FROM HERE?

Researchers such as Sfard and Prusak (2005) and Cobb et al. (2009) are interested in a

formalization of research on student identity. My own views about attempts to use

tools in prescriptive manners to gain insight into the individuals perceived identity

leads to difficulties: 1) The researcher will impose their own experiences onto the

experiences of the subjects, thereby reconstructing identity within a context that is not

purely specific to the student; 2) The student will be less likely willing to participate in

research if they believe that their own identity is erroneously represented.

As I further my own understanding of identity in mathematics education I am curious

about the identity students, as a collective, attach to mathematics. A deeper exploration

of the identity that exists in the boundary where students move to deeper affinity with

mathematics is fundamental to further my own ideas. Despite the cautionary warnings

I have with respect to the application of verbal indicators of identity outlined by Sfard

and Prusak (2005) I feel the need to utilize them in an analysis of my own data and then

re-evaluate their effectiveness and what I view they offer to the field.

As I begin to explore the language contained within the student responses to the

preliminary data, I will seek commonalities and possible insight into the identity that

students have assigned to mathematics.

References

Burton, L. (1994). Whose culture includes mathematics? In S. Lerman (Ed.), Cultural

Perspectives on the Mathematics Classroom (pp. 69-83). Boston: Kluwer Academic

Publishers.

Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the

identities that students develop in mathematics classrooms. Journal for Research in

Mathematics Education, 40(1), 40–68.

D’Ambrosio, U. (1997). Forward. In A.B. Powell & M. Frankenstein (Eds.),

Ethnomathematics: Challenging eurocentrism in mathematics education (pp. xv-xxi).

Albany, NY: SUNY.

Gee, J.P. (2001). Identity as an analytic lens for research in education. Review of Research in

Education, 25, 99-125.

Geertz, C. (1983). Local Knowledge: further essays in interpretive anthropology. Basic

Books, United States.

Grootenboer, P. (2013). The praxis of mathematics teaching: developing mathematical

identities. Pedagogy, Culture & Society, 21(2), 321-342.

Nasir, N. S. & Cooks, J. (2009). Becoming a hurdler : How learning settings afford identities.

Anthropology & Education Quarterly, 40(1), 41–61.

doi:10.1111/j.1548-1492.2009.01027.x.41

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Nixon, M. (1994) The culture of the mathematics classroom an unknown quantity. In S.

Lerman (ed.), Cultural Perspectives on the Mathematics Classroom (pp. 7-35). Boston:

Kluwer Academic Publishers.

Restivo, S. (1993). The promethean task of bringing mathematics to earth. In S. Restivo, J.P.

Van Bendegem, & R. Fischer (Eds.). Math Worlds: philosophical and social studies of

mathematics and mathematics education (pp. 3-17). Albany, NY: SUNY.

Restivo, S., Van Bendegem, J. P., & Fischer, R. (Eds.). (1993). Math worlds: Philosophical

and social studies of mathematics and mathematics education. Albany, NY: SUNY.

Sfard, A. & Prusak, A. (2005). Telling identities: in search of an analytic tool for investigating

learning as a culturally shaped activity. Educational Researcher, 34(4), 14-22.

MEDS-C 2015 Parhizgar, Alamolhodaei

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STUDENTS’ FLOW EXPERIENCES IN THREE TYPES OF

MATHEMATICAL PROBLEMS AND TWO DIFFERENT

EDUCATIONAL ENVIRONMENTS

Zakieh Parhizgar Hassan Alamolhodaei

University of Mashhad University of Mashhad

In this study, flow experiences of 244 students in three types of mathematical problems

(intra-mathematical problems, word problems and modelling problems) were

examined. Pre-test and post-test groups were used in this study. The treatment was that

the students, in two groups with different training formats (teacher-centered and

student-centered), attended 6 sessions of modelling problem solving. The results of this

study show that students’ flow experiences don’t have a significant difference in these

three types of mathematical problems. Teaching modelling problems increased

students’ flow experiences in reality-related mathematical problems, but it didn’t have

any effect on students’ flow experience in intra-mathematical problems. Also, the

student-centered group had positive effects on flow experiences in word problems.

INTRODUCTION

English and Sriraman (2010) emphasize that “the more we can incorporate genuinely

real-world problems within the curriculum, the better our chances of enhancing

students’ motivation and competencies in mathematical problem solving” (p. 268).

So, it is expected that by providing activities which has a reference to reality we will be

able to strengthen the learning process, improve students’ motivation, activate positive

emotions, and increase the number of students who enjoy mathematics (Schukajlow et

al., 2012). Many students are not motivated to learn mathematics and that is because

they believe mathematics is not related to the real world and they are often anxious and

impatient while practicing it. Do they like modelling problems more and involve more

in problem-solving process?

THEORETICAL BACKGROUND

Flow theory

A useful model for evaluating subjective experiences while teaching specific

assignments is the flow experience theory (Csikszentmihalyi, 1990). Csikszentmihalyi

(1990) describes flow experience as follows: A kind of favorable subjective experience

in which the person feels cognitively capable, has deep concentration on the task and is

highly motivated to continue it. Also, the person feels a pleasure which is derived from

the balance between the skills and challenges (Asakawa, 2004). Creating appropriate

challenges and offering opportunities for enhancing the skills (e.g. providing

immediate feedback and the gradual teaching of the more complex skills based on the

previously learned ones) can be one of the most ideal ways for engaging and attracting

the students (Csikszentmihalyi, 1997). In this study, we examine the flow experiences


85

of students while solving mathematical modelling problems and compare them with

other types of mathematical problems.

Mathematical problems based on their connection or lack of connection to the

real world

Mathematical problems are often divided into three groups: modelling problems, word

problems, and intra-mathematical problems (Niss, Blum, & Galbraith, 2007, p. 12).

The following describes each of these problems: (1) Modelling problems.

Mathematical modelling is: applying mathematics in solving problems in real-life

situations which don’t have regular structure (Galbraith and Clatworthy, 1990). One of

the models that can be used for describing modelling activities is the modelling cycle

proposed by Blum and Leiss (2007). (2) Word problems. Word problems are nothing

more than unmasking a purely mathematical problem which is expressed in terms of

the real world (Blum and Niss, 1991). (3) Intra-mathematical problems. The third type

of the problems is the ones which does not have any connection to reality and are

introduced using mathematical propositions. For solving them, only the appropriate

mathematical procedures and concepts are needed.

RESEARCH QUESTIONS

This study has been designed and done to answer the following questions:

Question 1: In which type of mathematical problems (intra-mathematical, word,

and modelling) students experience more flow?

Question 2: What are the effects of teaching mathematical modelling problems

on students’ flow experiences on different mathematical problems?

Question 3: Are students’ flow experiences different in teacher-centered and

student-centered methods?

METHOD

This research was conducted as pre-test, educational intervention, and post-test. Two

different teaching methods were employed: the direct teaching method

(teacher-centred) and the operative strategic method (student-centred) (see Blum,

2011). Because some studies show that learning activities based on student-centered

teaching methods can result in increasing motivation and positive emotions in students

(see, Hänze & Berger, 2007; Gläser-Zikuda, Fuß, Laukenmann, Metz, & Randler,

2005). The current study was done in nine grade 10 classes. The participants of this

study were 244 female students aged 15-16 from three private high schools. The

majority of students in those schools had high scores in mathematics. The sample size

was estimated using PASS software, considering at least 80% power for t-test in

pretest and posttest independent sample t-test (its nonparametric equivalent, the

Wilcoxon test) and a significance level of 5% based on the result of Schukajlow et al.,

(2012) study. So, 72 students were in a student-centred classroom (3 class) during the

course of this study while 172 were in a teacher-centred classroom (6 class). The study


86

consisted of a series of three initial math tests that included tasks from all three

problem types. After completing the tests, the students completed a flow questionnaire.

Next were six teaching intervention sessions during which students received direct

instruction in modelling problems. After the intervention, the students again completed

three math tests that included all the problem types, followed by completing the same

flow questionnaire. In the intervention sessions students were trained using modelling

problems about “Pythagorean theorem” and “Linear Functions.”

This study was done by three teachers who were familiar with modelling problems

beforehand. Also, they were provided with all problems and solutions of the training

sessions. It should be noted that one of the teachers who participated in this study is the

author of this paper and 7 classes were run by her and the two other teachers were

responsible for carrying out the research through teacher-centered method. This

method was consistent with the classes, since most mathematics classes are conducted

in the traditional way in Iran.

Measuring Tools

In this study, the Flow Perceptions Questionnaire (Cronbach’s α = .82) substantiated

by Egbert (2003) with minor changes was used to examine the flow experiences. It

consists of 14 items, but one of them was not understandable for students after being

translated to Persian, so it was omitted from the items(During this task, I could make

decisions about what to study, how to study it, and/or with whom to study). The

Perceptions Questionnaire measures the quality of subjective experience such as level

of interest, degree of concentration, enjoyment of the activity, and amount of perceived

control of the activity and there are no items directly related to perceived balance

between challenge and skills, so two items related to the balance between challenges

and skills were added to the questionnaire(My mathematical skills were in par with the

provided challenges & I believe that my skills enabled me to overcome the challenges.

So the flow questionnaire that was used in this study consists 15 items (see appendix).

Respondents fill out the flow questionnaire immediately after they solve mathematical

problems in pre-test and post-test. Egbert used this questionnaire to find out whether

flow happens in foreign language classes (see Egbert, 2003). This questionnaire has

been used in other studies (Sedig, 2007; Azizi & Ghonsooly, 2015). This questionnaire

is based on Likert format, having a 5-point scale from 5 (very strongly agree) to 1 (very

strongly disagree). An example of one of the items is: This task excited my curiosity.

RESULTS

Data analysis

In this study, parametric tests (repeated measures ANOVA test type I and type II) were

used for answering the research questions. The major assumptions of repeated

measures ANOVA were examined: (1) the errors were normally distributed (2) the

error term epsilon had constant variance (3) the errors were uncorrelated at lag order

one. Reliability of the questionnaires was examined using Cronbach’s alpha test: see

Table 1.


87

Reliability (Cronbach’s alpha)

Post-test Pre-test

0.893 0.886 Intra-mathematical

problems

0.901 0.880 Word problems

0.935 0.918 Modelling problems

Table 1: Reliability of the flow scales

Type of task and students’ flow experiences

In order to examine the first question, the repeated measures ANOVA test type I has

been used. Statistical analysis shows that there is no significant difference in students’

flow experiences among these three types of problems (F(2,486)=2.9, P=.064);

however, their mean (see table 2) shows that the mean of flow experiences in word

problems are the highest and the intra-mathematical and modelling problems are in the

second and third place respectively.

The effect of teaching modelling problems on students’ flow experiences

For examining students’ flow experiences after teaching modelling problems, the

repeated measures ANOVA type II has been used. The results obtained from repeated

measures ANOVA type II test shows that teaching modelling problems does not affect

the flow experiences in intra-mathematical problems (F(1,242)=2.02, P=.157), has

positive effect on students’ flow experiences in word problems (F(1,242)=18.67,

P<.001) and modelling problems (F(1,242)=28.48, P<.001). Also, comparing the

mean of flow experiences in pre-test and post-test in table 2 shows that students’ flow

experiences have increased significantly.

Post-test

M (SD)

Pre-test

M (SD)

3.42 (0.68) 3.34 (0.67) Intra-mathematical

problems

3.55 (0.68) 3.40 (0.69) Word problems

3.51 (0.80) 3.28 (0.79) Modeling problems

244 244 Number of cases

Table 2: Students’ flow experiences at pre-test and post-test

The effect of teaching method on students’ flow experiences

Results obtained from repeated measures ANOVA type II test indicate that the effect

of teaching group on students’ flow experiences in intra-mathematical problems is not

significant (F(1,242)= 0.64, P=.425). However, it was expected that students’ flow


88

experiences in intra-mathematical problems do not alter in any of the groups since the

intervention was about solving modelling problems.

Teaching method did not have a significant effect on students’ flow experiences in

modelling problems (F(1,242)= 2.05, P=.153). Table 3 shows that the flow

experiences have increased in both groups, modelling problems were new to students

in both groups and that is why these problems were interesting enough and deeply

involved students in solving such problems. However, it can be observed that the mean

of flow in student-centered group is higher than the mean of flow in teacher-centered

group.

Directive Operative-strategic

Posttest

M (SD)

Pretest

M (SD)

Posttest

M (SD)

Pretest

M (SD)

3.46

(0.71)

3.35

(0.70)

3.34

(0.60)

3.31 (0.60) Intra-mathematical

problems

3.58

(0.74)

3.50

(0.68)

3.48

(0.53)

3.17 (0.65) Word problems

3.51

(0.79)

3.31

(0.79)

3.53

(0.82)

3.19 (0.80) Modeling

problems

172 172 72 72 Number of cases

Table 3: Students’ flow experiences in the “operative-strategic” and “directive” forms

of teaching

The teaching group affected the flow experiences only in word problems. Specially, by

comparing the mean of students’ flow of word problems in pre-test and post-test (Table

3), we can observe that students experienced more flow in student-centered group

(F(1,242)= 6.42, P=.012).


The main goal of the current study was to determine in which kind of mathematical

problems students experience more flow. In general, students’ flow experiences such

as perceived balance between challenges and skills, level of interest, degree of

concentration, enjoying the activity, and received control over the activity have been

examined. Involvement in solving three types of mathematical problems based on their

connection or lack of connection to reality is the activity which was considered for

students.

The current study found that students could not differentiate between these three types

of mathematical problems before the intervention in fact, students’ motivation and

interest were the same while solving modelling, word and intra- mathematical

problems, but after it they experienced more flow while solving word problems.

Statistical analysis (repeated measures ANOVA test type I) shows that there is a


89

significant difference in students’ flow score among the three types of mathematical

problems (F (2,486)=3.61, P=.030). Table 2 shows the mean of students’ flow

experiences in pre-test and post-test. The mean of flow experiences in word problems

are the highest in both of them. Another study comparing enjoyment related to

intra-mathematical problems versus word problems (Frenzel et al., 2006; Pekrun et al.,

2007) found that students enjoyed word problems more than intra-mathematical

problems.

Pre-teaching modelling problems in order to improve students’ performance on these

problems and their familiarity with them had positive effects on increasing flow

experiences in both word and modelling problems, specially, word problems among

other types of mathematical problems. These results may partly be explained by the

fact that word problems are both connected to the real word and easy for students to be

motivated for solving them.

Another result of this study is students’ flow experiences in solving intra-mathematical

problems was not significantly different. Pre-teaching was related to mathematical

modelling problems, hence, any increase in flow experiences in intra-mathematical

problems is not expected.

This paper has compared different methods for teaching mathematical modelling

problems. The teaching method affected the flow experiences only in word problems.

An implication of enhancement of students’ engagement while solving word problems

in student-centered group is the possibility that the cooperative learning environment

create positive emotions in students.

The impact of students’ perceived quality of teaching on flow experiences is obvious.

Student-centred method had more positive effects on flow experiences. According to

the teachers’ observations, most of the students in teacher-centred group tended to

work in groups because of difficulty of modelling tasks. In general, therefore, it seems

that cooperative learning environments have the potential for enhancing positive

emotions in students (Panitz, 2000), since scholars have long debated the modelling

problems are designed to solve in the small groups (Lesh and Zawojewski 2007).

This study was limited by the use of only two concepts of “Linear Functions” and

“Pythagorean Theorem” from chapter 6 and 7 of Iran Math textbook, grade 9. The

major limitation of this study is gender distribution of the sample. So, for

generalizability carrying out more research in different grades, with males, and with

other mathematical concepts is recommended. Additionally, the participants in this

study were chosen from private high schools that had high math scores so it would be

interesting to examine results of similar research carried out in public schools.


90

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123–154.

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argumentative genre. Journal of Language Teaching and Research, 6(1), 210-215.

Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling,

application, and links to other subjects-state, trends, and issues in mathematics instruction.

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Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical

modelling problems? The example “Sugarloaf” and the DISUM Project. In C. Haines, P.

L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA12) —

Education, engineering and economics (pp. 222 – 231). Horwood: Chichester.

Csikszentmihalyi, M. (1997). “Flow and Education.” The NAMTA Journal 22(2), 3–35.

Egbert, J. (2003). A study of flow theory in the foreign language classroom. The Modern

Language Journal, 87(4), 499–518.

English, L. D., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman &

L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp.

263-285). Advances in Mathematics Education, Series: Springer.

Frenzel, A. C., Pekrun, R., & Goetz, T. (2007). Perceived learning environment and students’

emotional experiences: A multilevel analysis of mathematics classrooms. Learning and

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Galbraith P., & Clatworthy, N. (1990). Beyond standard models-meeting the challenge of

modeling. Educational Studies in Mathematics, 21(2), 137-163.

Gläser-Zikuda, M., Fuß, S., Laukenmann, M., Metz, K., & Randler, C. (2005). Promoting

students’ emotions and achievement — Instructional design and evaluation of the ECOLE

approach. Learning and Instruction, 15(5), 481 – 495.

Hänze, M., & Berger, R. (2007). Cooperative learning, motivational effects, and student

characteristics: An experimental study comparing cooperative learning and direct

instruction in 12th grade physics classes. Learning and Instruction, 17(1), 29 – 41.

Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.), The

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“THE OTHER TEN”: AUDEN’S SENSE OF NUMBER

Sheree Rodney


This paper presents a case study of a student (age 5 years 6 months) name Auden, who

interacted with a touchscreen App called TouchCounts. This App was designed to

support children’s activities around counting. I use Sfard’s commognitive framework

to show how Auden thinks and learns about numbers. I show how Auden’s exploration

of numbers helps me understand the challenges children face when moving from

identifying subsequent number as they appear in the natural counting sequence to

identifying numbers that appears before and after each other.

INTRODUCTION

In this paper, I examine how the use of technology along with Sfard’s (2008)

commognitive framework, provides a window on how one student, Auden, thinks

about what it means to count. I first explain commognition, how it is extended and

applied to an episode in which Auden interacts with mathematics involving the use of

TouchCounts (Sinclair & Jackiw 2014) and then provide a brief description of the

functionalities of TouchCounts. Auden’s engagement is not limited to a student/tool

interaction but also involves exchange with peers and adults.

BACKGROUND

In my home country (Jamaica), children are deemed “numerate” as soon as they can

count from one to ten and up to twenty-five for those who are classified as “gifted”. As

a result, students enter the early childhood level of education as early as three years.

The criteria for entry at this age are being able to count up to ten (numerate) and to

recite the letters of the alphabet (literate). No doubt most, if not all, of our students can

memorize and repeat number words and match them to objects at an early age—an

aspect of their educational life which is left up to their parents. With this experience I

often wondered how it is that this crucial aspect of children’s number sense

development is left up to parents to take care of. More surprisingly, I wondered about

the effects this unconventional mode of building the foundation of number sense

would have on children’s further study of mathematics. This motivated me to conduct

research on the ways children think about counting.

LEARNING ABOUT COUNTING

Bermejo (1996) describes counting as “a complex ability [...] In fact, children need two

years or more to count correctly and probably even more time to use counting

competently” (p. 263). Surely, if counting is seen as “a complex ability” it is more than

likely that teachers may not spend enough time to ensure that the concept is adequately

developed, and highly unlikely that children will demonstrate secure foundation for

further mathematics. Gelman & Gallistel (1978) outlines five counting principles they

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claim are essential to how children develop counting skills. These are the one – to – one

corresepondence principle, the order irrelevance principle, the fixed order principle,

the abstraction principle and the cardinal principle. From these five principles students

are expected to count every object once and only once, recognize that there is no

difference in which the order of counting is done and the number words must be

produced in a constant order. In addition, children should also recognize that it is

insignificant whether or not the elements counted are identical and that the last number

counted represents the number of items in a set respectively. Gelman & Gallistel

(1978) claim is that if a child lacks understanding of any of these principle (s) it is more

than likely they will not be able to count correctly. The focus of this paper therefore, is

on ordinal ability rather than cardinality because this is often times overlooked and it is

also the emphasis of the “enumerating world” in which Auden was engaging. Coles

(2014), in making the distinction between the two (ordinality and cardinality), refers to

ordinality as “the capacity to place numbers in sequence” that is, to know which

number comes before or proceed another. In demonstating this, Coles highlights

Gattegno’s (1994) perspective of ordinality as “primary” to number formation, where

numbers are introduced as relations rather than denoting objects. As I will show, it was

this relational connection that was missing from Audens’ thought processes.

In combining these ideas counting, for Auden, could be seen as a complex ability

which requires actions on all five counting principles in order to identify the

relationship between numbers. This epitomizes Auden’s experience with counting as I

observed that though he is able to call the number names initially he is unable to show

that ten appears after nine and he relies on the aural number names to make decisions

about the relative placement of numbers. His journey through the interaction with his

peers and task shows complexity in terms of how he thinks about counting and

suggests that counting acts, for him, are an imitation of chanting number words. The

aim of this paper, therefore, is to explore whether or not Auden’s competence in

counting indicates an understanding of what counting entails.

TECHNOLOGY IN THE MATHEMATICS CLASSROOM

The widespread use of technology in the past decade has drastically changed

education. As such, the challenge is for educators and researchers to create appropriate

opportunities that will utilize and demonstrate effective use of them. Noss & Hoyles

(1996), in their discussion on computers and educational culture, suggest that there is

more power to technology in mathematics education today than its physical presence in

the learning situation. Their claim is that well-designed digital technology can provide

learners with a means of expressing mathematical ideas and helps to unravels ways of

thinking mathematically along with fostering discursive interactions with children. My

focus in this paper is to show how the affordances of TouchCounts helped to unravel

ways in which Auden speaks about counting and numbers in general.

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TouchCounts

TouchCounts (Sinclair & Jackiw 2014) is an iPad application that introduces counting

and adding to early childhood learners. Within TouchCounts, there are two

microworlds called the enumerating world (for counting) and the operating world (for

adding and subtracting). In the former world, learners use their fingers to tap on the

screen which produces numbered yellow discs as well as aural number names. In the

latter world tapping creates large circles (called herds) representing the cardinality of

the set produced by the fingers. One of the main features of TouchCounts is the

one-to-one correspondence where every touch on the screen represents only one

object. In this paper the emphasis is on the enumerating world because the task

demonstrated involves Auden’s experience with the ordinal aspect of counting. The

figures below (Figures 1 & 2) show the screen of TouchCounts when the children are

working in the enumerating world. Figure 1 shows the horizontal bar called the shelf

and a yellow disc with ten on the shelf. This was the task Auden and his peers were

asked to perform. To do this, children will tap nine times on the screen below the shelf

and once above the shelf. The disc of numbers representing nine taps falls from the

shelf once they appear with their number sound because the program was set on

“gravity on”. Figure 2 shows what the screen looks like when the gravity is off after

children tap ten times on the screen.

Figure 1: The tenth tap above the shelf (gravity on) Figure 2: Ten taps (gravity off)

THEORETICAL PERSPECTIVE

My goal in this paper is to better understand the way children think about numbers and

about counting in particular. To do so, I draw on Sfard’s (2008) commognition theory.

Commognition is a discursive approach with its basic foundation laid on the premise

that thinking is individualized communication and that mathematical activity is

discursive. Sfard also believes that when learning takes place the mathematical

discourse (a multimodal communicational activity) is changed. In Sfard’s framework,

mathematical discourse has four main characteristics: word use (what are the words

used to describe mathematics objects?); visual mediators (what artefacts are used as a

moderator for communication in the discourse?); routines (what meta-discursive rules

are used to navigate the flow of communication?); and, narratives (what text are

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presented that can be accepted as true within the mathematics paradigm; these can be

axioms, definitions and theorems?).

In the case of TouchCounts the visual mediators include the discs, the herds of

numbers, gestures and fingers that children use to communicate within the

mathematical activity and in a Sfards’ framework visual realization of a signifier is

also seen as visual mediators. Realization refers to the “perceptually accessible object

that may be operated upon in the attempt to produce a description of the object” (p.

154) while a signifier refers to “words or symbols that function as a noun in an

utterance” (p. 154). In other words, a signifier- realization relationship is the mediation

between the signifier (in this case ten) and the object signified by its realization (such

as ten counters). My interest, however, for this paper, is centred on word use, visual

mediators and routines because children at this age may not be participating in

‘endorsed’ narratives. I also adopt Sfard’s (2009) idea on gestural communication,

which is used as a visual mediator. Her claim is that gestures ensure that all

interlocutors “speak about the same mathematical object” (p. 197). I also draw on

Châtelet’s view of, “gestures as being pivotal sources of mathematical meaning”

(quoted in de Freitas & Sinclair, 2014, p. 64). Though Sfard did not speak specifically

about aural mediators, it was evident that the sound of number words plays an essential

role in how Auden thinks about counting and numbers in general. He could hear and

repeat numbers but was uncertain of what they meant. I therefore incorporated aural

mediators (analogously to visual mediators) as a characteristic of discourse to analyse,

with the intention to show that sounds are routine prompts whose presence increases

the likelihood of enactment of a routine.

METHOD OF RESEARCH

For 8 months I had the opportunity to interact with Auden at an after – school daycare

for k-2 children with ages ranging from 5 – 8 years old in Burnaby. Auden is a 5 year

6 months old male student with high levels of interest to learn. I worked along with two

other researchers, Nathalie and Sean for approximately one hour each week (30

minutes per group) with students in groups of 2 to 4. Their general custom after school

is to have snack then play while they wait for parents. It was during their play time that

we interacted with them using TouchCounts and a myriad of mathematical tasks in a

small room adjacent to the common area. The episode in this paper was taken from an

interaction with Auden, two students Whyles and Misha (pseudonyms) where they

were asked to perform the task of putting ten only on the shelf in the enumerating

world of TouchCounts.

Selection of children to participate in small group interactions was done in two ways.

Students were randomly selected either from a list presented by one of the educators or

when they walked into the research site volunteering themselves to “play”. They were

also purposely selected when the team of researchers recognized the need for some

participants to be given more opportunities to interact with TouchCounts. Auden had

attended more sessions than other students because he volunteered himself sometimes.

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He was particularly chosen for this research because of his unusual yet stimulating way

of talking about counting and numbers.

The session was videorecorded for approximately 30 minutes while students engaged

in task ranging from putting 10 on the shelf, to skip counting, adding and subtracting. I

viewed the video and paid keen attention to how Auden communicated about specific

tasks. The video was then analysed taking into account his word use and the role of the

visual and aural mediators as well as how routines were developed. In the following

episode, I present a transcript in which Auden tries to put ten on the shelf (to tap below

the horizontal bar nine times and then above the bar once) during his first encounter

with TouchCounts.

DATA PRESENTATION

The data presented here is a transcribed episode of Auden’s engagement with the

researchers, his peers and TouchCounts. It shows a portion of his journey in

understanding the relational connection between numbers and ultimately counting. I

outlined Auden’s experience and then show how commognition was used as a lens of

analysis to help me understand how he thinks about counting.

Auden’s experience with counting

In the beginning, Auden’s ability to count seems secure. When asked to count to ten, he

is able to call the number words from one to ten with competence and confidence. This

plays an important role in his discursive fluency in two ways. Firstly, it allows him to

identify the numerals as they appear on the screen and secondly, he is able to use these

routines each time he tries to get ten on the shelf.

The children are asked to explore TouchCounts to see what happens on the screen and

the episode below illustrates their first encounter.

14 A: thirteen, fourteen, fifteen, sixteen, seventeen, eighteen,…. twenty-six [counting (listening to the sound and watching the screen) while Misha is tapping]

15 R: How high do you want to go?

16 M: to...to…to …to a lot.

17 A: to a lot?...that is not even a number [opens both hands wide]

After a few seconds had passed Nathalie ask Auden to put only 10 on the shelf.

34 A: you said ten?

35 R: Yeah I did say ten

36 A: ten as in [leaving seven on the screen]

37 R: ten comes after seven

38 A: the one with the slanting line and one straight line …oh that ten? [Using finger to trace seven in the air]

39 W: that ten! [Tracing ten in the air with his finger 1 and then 0]

A few seconds after

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51 A: oh! That ten

52 R: That ten! yeah

53 A: the other ten that looks like the same ten…this ten and the other ten [holding up two fingers indicating that there is another 10]

54 R: can you try to get this ten? (Pointing to 1 and 0)

Figure 3 Figure 4 Figure 5 Figure 6

Word use and gestures

I am cognizant that gestures can be interpreted as being visual mediators, but I have

decided to incorporate gestures with word use because they usually complement a

mathematical word used in an utterance. The vocabulary used by both the children and

the researchers shows their thinking about numbers and constitutes words such as “a

big number”, “a bit less” and “a lot”. The children were asked how high they wanted to

go while exploring what the application can do and, interestingly, none of them gave a

definitive response. Instead, they suggest a “big number” or “a bit less” as Whyles is

demonstrating with the gesture shown in Figure 6. It was this gesture that helped

Auden produce a number that is not big but at the same time not small either

(twenty-eight). In this episode, Auden’s gesture in Figure 4, which accompanies the

word ‘a lot’, provides some insight into how he thinks about numbers, as being a range

that can be of varying size. Also, while he repeats “a lot is not even a number” [17],

showing that Auden went through an intrapersonal commognitive conflict; while he

knows ‘a lot’ is not a number, he shows by his gesture, an internalization of ‘a lot’ as a

large number. Auden continues along this path of intrapersonal commognitive conflict

when he speaks about “that ten” [51] after he was asked by Nathalie to put ten on the

shelf. He was unsure whether he is to put 10 on the shelf or 7.

Visual mediators

Auden, Whyles and Misha are able to communicate effectively about putting ten on the

shelf because they are able to see the yellow disc as they appear on the iPad screen. The

children are able to make decisions about when to tap above or below the shelf. Also

Whyles’ gesture of forming ten with his fingers (see Figure 5) helped Auden form the

realization of 10 as a symbol but not as the number of times to tap on the screen: he

uttered, “Oh! That ten” [51], but continued to attempt getting seven on the shelf

nonetheless. Auden does not reify (interchanging discourse about a process with

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discourse about objects) numbers as things with properties. In other words, he sees ten

as an object and not as the number that comes after nine. As a result of this he forms

two different visual-verbal realizations of the signifier ‘ten’ and, as such, he confuses

the symbol 7 for 10 [38]. He describes ten as the one with the slanting line and one

straight line, as seen in Figure 3. He was in a deadlock to decide which of these

symbols represent tapping ten times on the screen, and found it difficult to complete

the task of putting ten on the shelf.

Routines

Auden cautiously uses one finger to tap the iPad screen at every attempt to put ten on

the shelf. He repeats the same actions (touching the screen and listening for the

number word) in the same way each time he performs the task. The sound of the

number words, which is a part of the TouchCounts application, increases the repetitive

pattern of performing the task. Auden knows the routine of tapping several times above

the shelf and then once but was unclear of how many times to tap below the shelf. In

addition, routines appear in word use in the logical sequence as shown below:

# R: how high do you want to go?

# M: to a lot

# R: Can you get a big number?

# W: A big number, but a bit less

Routines also appear in the way the children took turns in performing the task,

allowing them to imitate each other.

Aural mediators

Auden relies heavily on the sounds of the number words in order to decide when to tap

above the shelf. In addition, number words appear mainly in ritual number chanting for

Auden, who acts upon the aural mediator as a routine prompt as the only means of

indicating some internalization of counting [14]. When the group was asked “do you

know what will come next?” Auden quickly tapped the screen to hear the number

name. He is minimally reliant on what is seen on the screen rather than what is heard.

As a result, he is unable to determine when to tap above the shelf and in many cases,

placing just ten on the shelf was futile. He would exclaim, “I missed it!” or Nathalie

would say “ten went away”


The analysis provides supporting evidence that Auden utilizes gestures, visual and

aural mediators along with a unique language to think and learn about counting.

Though Auden is able to demonstrate a stable order in counting initially, he faces a

predicament to realize that ten comes after nine, hence his inability to recognize and

utilize the ordinal aspect of counting. The analysis also shows that Auden is unable to

differentiate the symbols of ten and seven and is unsure about what is meant to get only

ten on the shelf. The difficulty that this analysis reveals is that repetition through

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memorization and number chanting does not induce an understanding of counting,

neither does it help children to identify the relational aspect of numbers suggested by

Gattegno (1974); as in this case where Auden relies heavily on the utterances of his

peers and the feedback of TouchCounts in the interaction. Using Sfard’s framework

helps me see the warning signs of ways in which non-learning may be sheltered when

students are engaged in traditional ways of teaching. It brought to the fore the

assumptions we make as mathematics educators of the projected and unprojected

activities of students as they engage with mathematics.

References

Bermejo, V. (1996). Cardinality development and counting. Development Psychology, 32(2),

263.

de Freitas, E & Sinclair, N. (2014). Mathematics and the body: Mathematics entanglements

in the classroom. Cambridge University Press.

Coles, A. (2014). Ordinality, neuro-science and the early learning of number. In P. Liljedahl,

C. Nicol, S. oesterle, & D. Allan (Eds.). Proceedings of the Joint Meeting of PME 38 and

PMENA 36 (Vol. 2, pp. 329-336). Vancouver, Canada: PME.

Gattegno, C. (1974). The common sense of teaching mathematics. New York: Educational

Solution Worldwide Inc.

Gelman, R. & Gallistel, C. R. (1978). The child’s understanding of number. Harvard

University Press.

Laborde, C. (2001). Integration of technology in the design of geometry tasks with cabri –

geometry. International Journal of Computers for Mathematics Learning, 6(3), 283–317.

Noss, R & Hoyles, C. (1996). Windows on mathematics meaning: Learning cultures and

computers. Mathematics Education Library. Kluwer academic publishers.

Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses,

and mathematizing. Cambridge University Press.

Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in

Mathematics, 70(2), 191 -200.

Sinclair, N. & Jackiw, N. (2014). TouchCounts [iPad application software].

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TEACHER TENSIONS: THE CASE OF NAOMI

Annette Rouleau


Tensions are endemic to the teaching profession. Viewed as dichotomous forces,

tensions shape the experiences of mathematics teachers, affecting both their practice

and professional growth. In this article, I use Berry's (2007) framework to identify and

examine some of the tensions experienced by Naomi in her practice of teaching

mathematics. While previous research presents the image of teachers as dilemma

managers who accept and cope with continuing tensions, my research suggests that a

desire to resolve these tensions may impact teaching practice and professional growth

needs.

INTRODUCTION AND THEORETICAL BACKGROUND

Teachers are often faced with dilemmas. Lampert (1985) suggests that these dilemmas

arise because the state of affairs in the classroom is not what the teacher wants it to be.

Conflicts can surface when mathematics teachers encounter a disparity between what

they want to do and what they are asked to do, or between what they want to do and

what they know how to do. These competing influences create what Adler (2001),

Berry (2007), and Lampert (1985) refer to as teacher tension, and encompass the inner

turmoil teachers experience when faced with contradictory alternatives for which there

are no clear answers. For both Adler (2001) and Lampert (1985), these tensions are

seen as problems to be managed rather than solved. Considering that tensions are

endemic to the teaching profession, mathematics education would benefit from (1)

identifying these tensions, (2) understanding how teachers cope with these

dichotomous forces, and (3) examining how this impacts their teaching practice and

professional growth requirements. While a longer version of this paper, addressing all

three areas, has been submitted to MAVI 2015, the goal of this particular paper is to

explore a framework for identifying and examining these tensions, in order to better

understand the wants and needs of mathematics teachers.

Whether categorized as personal, practical, or contextual, identifying tensions within

the practice of mathematics is beneficial in providing a language for discourse (Adler,

1998; Ball, 1993; de Oliveria and Barbosa, 2008). Indeed, Adler (1998) refers to the

language of tensions as "a powerful explanatory and analytic tool, and a source of

praxis for mathematics teachers." (p. 26). Naming and exploring the tensions can

present a view of teacher thinking that is broader than decision making (Lampert,

1986). To highlight this, Lampert (1985) shares an illustrative example of the tension

she experienced upon choosing where to sit her students during mathematics lessons.

No matter which arrangement she chose, it would be to the detriment of some of her

students. Outwardly appearing as a simple 'decision', the thought process entailed in

managing her tension demonstrates the complexity involved.

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A framework to understand the tensions inherent in teaching emerged from the work of

Berry (2007). As a former teacher who moved into the role of a teacher educator, she

completed a self-study of her practice in order to improve her understanding of the

process of learning to teach teachers. Building upon the work of Adler (2001) and

Lampert (1985), she utilized the notion of tension as a framework for both doing and

understanding her research. The result was twelve tensions expressed as dichotomous

pairs that "capture the sense of conflicting purpose and ambiguity held within each"

(Berry, 2007, p. 120). Noting that these tensions do not exist in isolation, she used their

interconnectedness as a lens to examine her practice:

1. Telling and growth

● between informing and creating opportunities to reflect and self-direct ● between acknowledging prospective teachers' needs and concerns and

challenging them to grow.

2. Confidence and uncertainty

● between making explicit the complexities and messiness of teaching and

helping prospective teachers feel confident to progress ● between exposing vulnerability as a teacher educator and maintaining

prospective teachers' confidence in the teacher educator as a leader.

3. Action and intent

● between working towards a particular ideal and jeopardising that ideal by

the approach chosen to attain it.

4. Safety and challenge

● between a constructive learning experience and an uncomfortable

learning experience.

5. Valuing and reconstructing experience

● between helping students recognise the 'authority of their experience' and

helping them to see that there is more to teaching than simply acquiring

experience.

6. Planning and being responsive

● between planning for learning and responding to learning opportunities as

they arise in practice (Berry, 2007, p. 32-33).

Although initially applied to teacher education, it is possible the tensions that emerged

from Berry's (2007) framework can be used both as a way to identify the competing

conflicts experienced by mathematics teachers, and as a way to describe them. As

such, my research question is aimed at identifying similar tension pairs within teachers'

practice of teaching mathematics. While my eventual goal is to explore how these

dichotomous forces impact teaching practice and professional growth needs, in this

study my purpose is only in the applicability of using Berry's framework to emerge sets

of tensions from the practice of a mathematics teacher. In what follows the

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methodology is addressed, one particular case is analyzed, and the results and

conclusions are discussed.

METHODOLOGY

This paper is part of an ongoing research project in which I will examine the tensions

of teachers from Kindergarten to University. In the end I will have data from 25

participants (5 primary, 5 intermediate, 5 junior high school, 5 senior high school, and

5 University teachers). In what follows I present a brief analysis of one of my first

participants, a teacher named Naomi.

Naomi has been teaching for eleven years: one year teaching grade five on a remote

First Nation reserve, three years teaching grade four in an urban K-8 school followed

by seven years teaching grade six in the same school. Her undergraduate degree

required her to complete one introductory mathematics course and one mathematics

for education course. A self-described 'mathematics-phobe8', she has had no further

mathematics education other than occasional mandated professional development.

Data was collected over a one-year period during which Naomi was a participant in a

District Learning Team led by myself. Notes were kept of conversations with Naomi

that occurred naturally during breaks in the sessions. The opportunity was taken

during these casual interactions to probe more deeply into questions Naomi had asked

or about perspectives she had shared. Field notes were also taken during two

classroom observations and during the lesson debrief. More formally, Naomi was

engaged in two follow up interviews that were designed to illuminate tensions present

in her practice of mathematics. These interviews ranged from 20 to 45 minutes in

length and were transcribed in their entirety. The data collected were then scrutinized

using Berry's (2007) framework as an a priori frame for identifying and coding

tensions. Dichotomous tensions in Naomi's practice emerged from the data and were

identified and categorized accordingly.

ANALYSIS

In the following analysis, Berry's (2007) framework will be used to analyze data from

the perspective of Naomi in her practice of teaching sixth grade children. For the

purposes of brevity, only three of the tensions will be discussed here.

Telling and Growth

For Naomi, the teaching of mathematics has been influenced by the constructivist

notion of learning. She understands this to mean that the primary role of teaching is not

to lecture, explain, or otherwise transmit mathematical knowledge, but to create

opportunities for the student to construct their own knowledge. Naomi wants to avoid

'telling' and instead focus on growth of understanding through experience. She

explains that her teaching style is "like night and day" in comparison with her own

mathematical education, which she described as rote memory, drill and kill, and

8 A ‘mathematics-phobe’ describes a person who has an aversion to mathematics.

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algorithms. Instead, she uses "hands-on, collaborative activities where students talk

and work through things together to come up with many different ways to do things".

Naomi values when students come to her with a new mathematical understanding that

she has not taught. Yet a tension arises when she is faced with students who are not

accustomed to being taught in the "new style of teaching". She shares that she doesn't

know how to deal with that, and sometimes reverts back to "the old school" despite

believing that it's really not in the best interests of the child. Naomi says she attempts to

help the students see the benefits of her teaching style but that there are days when she

hands out mathematics worksheets. Her statement, "and you don't want to do that, you

want to be egging these kids on to try and do as much as they can" reveals the tension

Naomi feels. It was apparent that this tension is a driving force in Naomi's professional

growth and development when she followed with "I'm always looking for better ways

to teach math". This is a tension that is neither managed nor resolved, instead it has

become the impetus for change.

Confidence and uncertainty

Teachers who feel weak in mathematics have the dilemma of whether or not to share

that weakness with their students. Berry (2007) suggests there's a tension between

exposing vulnerability and maintaining the respect of students, which Naomi

discovered when she first began teaching. Her unexpected answer to the question,

"What are you best at in teaching math?" was "Um, honestly, showing the kids my

weaknesses in math". Naomi goes on to reveal that mathematics is her weakest subject

and that she struggled with it throughout her life. When she began teaching, she hid this

from her students because she thought they expected her to know everything. Naomi

was able to accomplish this deception by a lot of traditional, direct teaching from the

text and passing over questions that she couldn't answer. This eventually felt

uncomfortable for her and she began telling her students when she didn't know

something, stating "But then I decided, maybe it would be okay if they knew". Using

the word 'maybe' in her rationale indicates that perhaps Naomi is not completely at ease

with her decision but she concludes with, "and I think that the kids actually love it even

more because the teacher really is struggling, and they get to teach me". Naomi has

made a decision that is acceptable to her and that she can live with. Her tension is not

resolved but it is managed.

Action and intent

Berry (2007) tells us that tension can arise when what we intend to do is in conflict

with our approach in working towards that intention. In other words, what we do can

inadvertently undermine our goals. Naomi provides an example of this tension when

she describes her reliance on summative assessments in mathematics stating, "I would

say that with math, my hardest struggle would be not to rely solely on summative

assessments." She notes that her teaching style requires the use of formative

assessment but she frequently forgets to document the students' learning and reverts

back to her traditional summative assessment adding, "I think because I get carried

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away in classes and I'm not actually documenting the formative assessments as I'm

going through." This causes tension for Naomi because her view of summative

assessment is that it doesn't provide an accurate reflection of what the kids are doing

"especially the kids that have text anxiety or they're having a bad day." Her intent

appears to be to assess the students in a way that best matches the way the content was

learned but her actions in relying on summative assessment interferes with that aim.

And, despite acknowledging that the results from the summative assessment

occasionally surprise her, Naomi manages this tension by combining the information

from the summative assessment with what she has observed in class. "It (summative

assessment) doesn't give a true reflection of what the kids are doing. You're looking at

summative assessment marks but also trying to reflect, bring in some of your

observations." Naomi shared that her decision to volunteer to be part of the District

Learning Team was in part because of her desire to learn more about assessment in

mathematics. Again, here is a tension in which seeking professional growth is seen as

part of the outcome in resolution.

DISCUSSION

In examining the data, it is possible to see limitations in applying Berry's (2007)

framework to a teacher/student situation. Further tensions are evident that do not

appear to fit within any of the categories provided by Berry (2007). Notably are the

tensions the participant experiences with parents and colleagues. Naomi willingly

spends time working with parents to help them understand "new math" but

acknowledges that "it takes a lot of my time and effort." With colleagues, Naomi

voices a tension between a latent desire to conform to "what everyone else is doing

because that's just easier" and doing what she believes to be pedagogically sound.

Teachers feel a great deal of pressure to conform to the norms and standards of their

school, their mentors, and their grade partners. While this is especially true for

beginning teachers, even experienced teachers feel tension when abiding by the norms

conflicts with personal pedagogical beliefs.

It would appear that, in order to encompass Naomi's tensions surrounding parents and

colleagues, new categories of tension pairs would be of possible benefit. I expect that

as my research progresses and broadens I will also be capturing other tensions that are

specific to the practice of teaching mathematics. This will require expanding or

perhaps altering Berry's (2007) original framework to allow for their incorporation.

It may also be beneficial to consider further categorizing the tensions according to

whether they are conflicts of pedagogy, conflicts of subject matter or possibly conflicts

from external, systemic influences. What information might be revealed that could

help in understanding how teachers experience tension? As well, on several occasions

Naomi mentioned the difference between her current practice and when she first began

teaching. It would be beneficial to compare the tensions felt by a new teacher with

those of an experienced teacher and then examine any differences in how they manage

those tensions.

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CONCLUSION

Naomi appears to fit Lampert's (1985) image of a teacher as a dilemma manager who

accepts and copes with continuing tension. She initially manages the tensions that

surface in her practice while never fully resolving her competing conflicts. Our results

show that where Naomi may differ is in living with the consequences of her decisions,

as some of her managed tensions continue to resurface. She acknowledges that she

wished she "knew how to deal with that". This results in new tensions between what

she wants to do and what she knows how to do. The new tensions Naomi experiences

may potentially fuel a desire for change in her teaching practice. In my future research

I hope to discover if this is applicable to other teachers' experiences of teaching

mathematics and what impact that might have on the design and delivery of

professional development. By further investigating teacher tension, I believe the field

will be better informed to improve teacher education and professional development

efforts related to this phenomenon. It is beneficial to mathematics education to have a

fuller understanding of these tensions that drive teachers' needs, and shape both the

individual and their practice.

References

Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht:

KluwerAcademic.

Adler, J. (1998). A language of teaching dilemmas: Unlocking the complex multilingual

secondary mathematics classroom. For the Learning of Mathematics, 18(1), 24-33.

Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching

elementary school mathematics. The Elementary School Journal, 373-397.

Berry, A. (2007). Tensions in teaching about teaching: Understanding practice as a teacher

educator. Dordrecht: Springer.

Barbosa, J. C., & de Oliveira, A. (2008). The teachers' tensions in the practice of

mathematical modelling. In Proceedings of the 11th International Congress on

Mathematical Education.

Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in

practice. Harvard Educational Review, 55(2), 178-195.

Lampert, M. (1986). Teachers' strategies for understanding and managing classroom

dilemmas. In Advances of Research on Teacher Thinking. Berwyn, PA: Swets North

America Inc.

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A BLIND UNDERGRADUATE STUDENT’S JOURNEY TOWARD

AN UNDERSTANDING OF PRE-CALCULUS CONCEPTS

Mina Sedaghatjou


In this paper we argue that although mathematical communication and learning are

inherent multimodal and embodied subjects, sight disabled students are also able to

conceptualize visuospatial information. Adapting Vygotsky’s mediation theory, we

show that the lack of access to visual fields in an advanced mathematics course does

not obstruct a blind student in visualization, but rather modifies it. We argue proper

precise tactile materials might empower blind students to better visualize

mathematical functions.

SIGHT DISABILITY ROUSTES OTHER ABILITIES

Euler (1707–1783) and Saunderson (1682–1739) are two well-known mathematicians

who had been struggling with blindness. Jackson, a contemporary blind mathematician

(2002), argues that visuospatial imagination amongst people who do not see with their

eyes relies on tactile and auditory activities, however mathematical instruction and

communication involves gesture, gaze, posture, pointing and facial expression, while

speaking (Quek, McNeill, & Oliveira, 2006). One of the central tasks for learning

mathematics in general and pre-calculus in particular, is to understand, visualize and

interpret mathematical concepts, graphs, and objects (Healy & Fernandes, 2011;

Healy, 2012). Healy (2014) claims that visualization goes beyond "seeing" and that it

can develop in the absence of vision. She explains that visualization consists of other

sensorial perceptions, relationships with previous experiences and knowledge,

verbalization, and more (Healy, 2014).

Mathematical and statistical graphs contain a concise, complete and precise summary

of functions and equations such as: employing ordered pairs, axes, origin, grid lines,

tick marks, intersections, labels, etc. Obviously, these components are all visual. As a

result, for students with vision impairment, understanding the mathematical concepts

behind them or learning the concepts themselves becomes an extremely challenging

task. Teachers working with visually impaired students struggle to convey the concept

of a graph to their students, specifically when the concept begins to get more advanced

and complicated. This suggests that to understand the learning processes of visually

impaired and blind mathematics learners, it is important to investigate how the

particular ways in which they access and process information shapes their

mathematical knowledge and the learning trajectories through which that knowledge is

attained. In the UNESCO’s World Conference on Special Needs Education (1994) in

Spain, the Salamanca Statement affirmed that there is the necessity and urgency of

providing education for children, youth and adults with special educational needs

within the regular education system.


107

For years, most of the students with vision disabilities avoided taking mathematics

courses at higher levels of education and tended to feel anxious and negative towards

math. As the result, avoiding taking math courses hindered them in moving toward

their desired fields of study, jobs and careers.

There are few studies on learning and teaching mathematics for learners with visual

impairments from early childhood to the secondary level; in addition, there is also

scant research focusing on university level students tackling understandings of

advanced mathematical concepts. To help redress this situation, in this paper which is a

part of larger research investigating the roles of creative haptic materials that empower

blind students to visualize advanced mathematical concepts.

A motivating example

Mathematics communication and conceptualization have strong embodied

components. Embodied communication, gesticulation, gaze, pointing, and

body-language in mathematics discourse has a critical role in communication between

sighted individuals discussing math concepts. In “The Emperor’s New Mind”,

mathematician and physicist Roger Penrose wrote:

Almost all my mathematical thinking is done visually and in terms of nonverbal concepts,

although the thoughts are quite often accompanied by inane and almost useless verbal

commentary, such as ‘that thing goes with that thing and that thing goes with that thing.

(Penrose, 1989, p. 424)

While the functionality of mathematically-grounded gesticulation involved with deixis

is huge, we show in the following episode that communicating in a way that is

understandable to all parties in teaching and learning mathematics, could provide

critical information to maintain mathematical understanding. In this paper, we are

seeking how embodiment could be seen as the visual cue for a blind learner in

mathematical communication. We also articulate that mediating resources make

mathematical communication possible for blind learners in advanced mathematics

courses.


We believe mathematics to be, at least in large part, a social and cultural product. It is

not an abstract, eternal and isolated subject out there, totally independent of human

cognition and understanding. Therefore, construction, representation and manipulation

of mathematical objects are variously invented and constructed through social and

cultural activities. Our framework accords with Wittgenstein and the Participationism

theory of Lave and Wenger (1991), which asserts that learning is a process when

students become the participant in a certain community and engage in specific

activities. So, for the blind learner many mathematical concepts that involve working

with spatial representations and information, his/her hands represent the most obvious

substitute for the eyes, and hence it is not surprising that research involving blind

geometry learners has focused on how explorations of tactile representation of


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geometrical objects contribute to the particular conceptions that emerge. For this study,

we adopt Vygotsky’s “mediation” theory (1986) with disabled learners, and will take a

qualitative perspective to explore how access to different mediating resources impacts

learning mathematics (Vygotsky, 1986).

METHODOLOGY

This study is part of a larger research project exploring “Issues And Aids for Teaching

and Learning Mathematics to Undergraduate Students with Visual Impairment”.

Prospective participants, who are visually impaired, are identified by the Center of

Students with Disabilities’ (CSD) specialists. The participant focused on for this paper

is a blind student named Anthony, who was taking a pre-calculus course.

Anthony, the learner described here, is a twenty-eight-year-old male, who is doing his

sixth year of undergraduate studies in kinesiology and health science at a Canadian

university. He was born with a profound visual disability and is completely blind. He

has passed “Mathematics Foundations” successfully the prior semester as the

prerequisite for enrolling in pre-calculus.

Anthony has excellent mental calculation skills, which enables him to easily do basic

calculations without a calculator. He is very keen to create a pathway for other

sight-disabled students to pursue their academic dreams by taking mathematics

courses. What follows is a brief description of material and software use and

procedures that we followed to assist Anthony through his pre-calculus journey.

Written materials

Braille, which is well-known as a tactile writing system designed for the blind and

visually impaired individuals, is a strictly linear notation and not generally useful for

mathematics and its various notations (Marcone, 2013). In addition, Braille readers can

only perceive what is under their fingers at the time, so it can be very difficult for them

to obtain a general view of algebraic expressions and graphs as a whole. Braille

extension specialized for mathematical notation known as the Nemeth Code has

different coding than Braille itself . Considering Braille and Nemeth code’s limitations,

we chose LaTex to translate mathematical language and formulas and prepare other

written materials. LaTex is used to prepare textbook content, class and lecture notes.

LaTex

The most important reason that we chose to adopt LaTex was that we found, LaTex a

common language that all parties (instructor, learner, tutor and Anthony’s assistant)

that were involved in teaching and learning the course could communicate with. A

table of contents at the top of each LaTex and Word file was used to give insight into

what is compiled in each file, so Anthony did not have to go through the whole

document to figure out what is in it. Anthony also suggested that descriptive file titles

are best for identification purposes rather than the date of the class. For example:

\frac{numerator}{denominator} uses for fractions, % for comments and all the


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mathematics formulas places between $s. So, for example,

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ means .

JAWS

Digital technologies are facilitating conversions between written text and blind

learners by offering spoken versions of written materials and mathematics equations.

Furthermore, use of spoken rather than written materials suggest that the ears can also

be used as alternates for the eyes. Having said the limitation and possibilities of

incorporating different senses for a blind learner, for conducting this research we

mostly rely on auditory and tactile materials.

JAWS9 (Job Access With Speech), is a computer screen reader program for Microsoft

Windows that allows blind and visually impaired users to read the screen either with a

text-to-speech output or by a Refreshable Braille display, was used by Anthony. This

program reads most of the programs and actions in Windows and LaTex. Anthony’s

laptop reads using JAWS. VoiceOver in Mac iOS also does the same action (Figure

2-A).

Tactile graphs: A real challenge that tackled

One of the very challenging problems here is to help a blind learner on visualizing

mathematical graphs and concepts. We face this difficulty in various forms, in class

and during lecture, and at home and doing exercises. To tackle this, when there was an

image or figure displayed on the overhead in the class and during lecture, or when

lecturer uses “this” and “that” for describing a mathematical figure, I (the first author)

rapidly made graphs using Net-board and crayons (Figure 1A& 1B) or Wikki Stix

(Figure 1C &2B).

Net-board is a wooden board with a metal net installed on it. As it is shown in the

graph, using crayons while paper is placed on the net-board creates a texture that is

tangible for Anthony. Making the graph on the net-board was fast and adequate to

provide a mental image of the graph to the learner.

9 JAWS is produced by the Blind and Low Vision Group of Freedom Scientific, St. Petersburg, Florida, USA.


110

A B C

Figure 4: A&B Innovative way for drawing graphs during lectures and tutoring using

“metal net-board”, crayons and normal papers. C: Examples of using Wikki Stix on a

raised line graph paper at Georgia Academy for the Blind

To embrace presenting textbook graphs, when there was enough time, I (first author)

created graphs using various materials and textures, trying to make it as precise as

possible. For this purpose, firstly axes and grids were printed on graph paper using a

Braille embosser10. Then, I drew graphs using hot glue, a wheel tracing tool, and

stickers. For labelling, I used Nemeth code on top of the graph paper, in a similar way

that represented the original source (Figure 2-C).

A B C

Figure 5- A: VoiceOver is reading class note. B: Anthony is creating graph of function

by Wikki Stix. C: Making various textures using different materials to make graphs

perceivable, and hence conceivable, as precisely as possible

Demonstrating function’s behaviour by the blind learner

In this section, we start with describing the participant’s action and communication,

when he was asked to describe the behaviour of for the sighted

interviewer. This episode happened after Anthony graphed the function using Wikki

Stix on raised grid X-Y axes (Figure 2B), which is out of this paper’s scope and we do

not report on it. Then, we analyse developing gestural, embodied and discourse

communication.

10 Braille embossers are "printers" for Braille.


111

Anthony taps on the quadrant two on the haptic graph created by the first author

(Figure 2-C), and says: “I'm in quadrant two, the function goes like this”. He moves his

left hand (illustrating function) from left to the right horizontally.

Anthony ...and hugs the vertical asymptote at y=1[ brings his right forearm parallel

and right below the left forearm], and as it gets closer to [pause]…the

Y-axes... [pause]...[left and right forearm moves upward] it goes up..

[pause- he seems not sure].

Then, Anthony touches the graph, and accordingly recognizes his explanation was not

correct. Then, he waves his left hand to emphasize that this will demonstrate function

and moves it from left to the right horizontally.

Anthony The function, goes horizontal, hugs up the horizontal asymptote at Y=1

[two forearms touch each other] (Figure 3A) and then, it goes up to the

vertical asymptote at x= -2[two hands moves upward, left wrist taps right

forearm] (Figure 3-C). That means that [function] doesn’t cross it [vertical

asymptote] and it comes up (Figure 3-B).

Subsequently, Anthony “looks at his notes” (Figure 3D) by touching haptic graph.

Anthony …And the middle [part of the] function, is where the function is between

the two vertical asymptotes [-2<x<3]; doesn’t cross either ones [waves his

forearms] (Figure 3E). But, it does cross upward, over the origin, and over

the horizontal asymptotes at y= 1 and comes toward to the vertical

asymptote at x=3 [ Taps with his right wrist(asymptote) to his left forearm

(function)], and again, never crosses it (Figure 3F). And this is middle part

of the function [ taps on the middle part of the graph]

Afterwards, Anthony taps in each quarter, while naming them. He explains the

behaviour of right side of the function, for x>3. He points and taps on his right hand to

stress that demonstrates the function and not asymptote (Figure 3G).

Anthony This is my function. It comes up along the vertical asymptote at x=3 (Figure

3L). And, as it comes up, instead of crossing over horizontal asymptote at

y=1, it gets close to it [bends both forearms, while they are sticked together.

Bends his body to the right], and does not cross (Figure 3M).

A B C


112

D E F

G L MM

Figure 3: Anthony explains function’s behaviour for , using his hands and

forearms and body. Each arrow shows the direction and path of movement (Red

dashed arrows: vertical asymptote, blue dashed line: horizontal asymptote and yellow

arrow: function)

Selected episode above presented how embodiment works as visual cue for a blind

learner. That also exemplified the modified multimodal communication in absence of

sight. Whereas, Anthony uses his finger tips to explore diverse information on graph

papers and other written resources, he does not use them, as sighted learners do.

Instead, other parts of his body such as his forearms and wrist, while wave, move,

touch, slide and tap demonstrates high level of body engagement and mathematical

embodiment. We have shown the capacity of the blind for visualization and spatial

reasoning (Quek, et al., 2008). In addition, the verbs that were used by Anthony (e.g.:

moves, hugs, cross etc.) displays that he percives mathematics entities as dynamic

objects.

CONCLUSION AND REMARKS

Although teaching and learning mathematics at the secondary level seems challenging

and time consuming for sight disabled learners, it is not impossible. We found a high

degree of sensuous and body engagement in understanding the pre-calculus concepts

by Anthony in other than visual modes. This is inline with neurological evidence to

support this position suggested by Gallese and Lakoff (2005) “that circuitry across

brain regions link different sensorial modalities “infusing each with the properties of

others” (Healy, 2012, p.5). We also found, as Vygotsky’s (1986) claims, providing

proper and timely semiotic or material tools “alters its entire structure and flow”, while

the learner is actively engaging in processes (Healy, 2012, p.2). In other words, those


113

activities could cover and modify visualization via other senses such as tactile and

auditory rather than visual. Future researches could reveal what kind of mediators and

how they could help the sight disabled learner to successfully pass other courses such

as advanced biology in university level. In addition, we found that teaching and

learning advance mathematics is very time consuming for visual disabled learners. So,

more studies are needed to figure out how university policies should be modified for

those learner.

References

Choi, S. H. (2010). Digitizer Auditory Grap h : Making Graphs Accessible to the Visually

Impaired, 3445–3450

Healy, L. (2012). Hands That See , Hands That Speak : Investigating Relationships Between

Sensory Activity , Forms Of Communicating And Mathematical Cognition.

Healy, L. (2014). Blind Students, Special Needs, and Mathematics Learning.

Healy, L., & Fernandes, S. H. A. A. (2011). The role of gestures in the mathematical practices

of those who do not see with their eyes. Educational Studies in Mathematics, 77(2-3),

157–174. doi:10.1007/s10649-010-9290-1

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. (R. Pea

& J. S. Brown, Eds.)Learning in doing (Vol. 95, p. 138). Cambridge University Press.

doi:10.2307/2804509

Marcone, R. (2013). Teaching Mathematics for blind students : a challenge at the university.

In ICME 12 (pp. 23–35). Seoul.

Penrose, R., The Emporer’s New Mind. 1989, New York: Oxford University Press.

Quek, F., McNeill, D., & Oliveira, F. (2006). Enabling Multimodal Communications for

Enhancing the Ability of Learning for the Visually Impaired. Proceedings of the 8th

International Conference on Multimodal Interfaces (ICMI), 326–332.

http://doi.org/10.1145/1180995.1181056

Vygotsky, L. (1986). Thought and Language. (A. Kozulin, Ed.)October (Vol. 28, p. 287).

MIT Press. doi:10.2307/1249570

http://doi.org/10.1145/1180995.1181056

MEDS-C 2015 Truman

114

MATHEMATICS LEARNING AMONG UNDERGRADUATES ON

THE AUTISM SPECTRUM

Jeffrey Truman


This study examines the mathematical learning of an undergraduate student on the

autism spectrum, as well as some views of autism in the Vygotskian framework. I aim

to expand on previous research, which often focuses on younger students in the K-12

school system. I have conducted a series of interviews with one student, recording

hour-long sessions each week. The interviews involved a combination of asking for the

interviewee's views on learning mathematics, self-reports of experiences (both directly

related to courses and not), and some particular mathematical tasks. I present some

preliminary findings from these interviews and ideas for further research.

BACKGROUND ON AUTISM-RELATED RESEARCH

The Autistic Self Advocacy Network (2014) states that autism is a neurological

difference with certain characteristics (which are not necessarily present in any given

individual on the autism spectrum), among them differences in sensory sensitivity and

experience, different ways of learning, particular focused interests (often referred to as

'special interests'), atypical movement, a need for particular routines, and difficulties in

typical language use and social interaction. Over the past few decades, there have been

many research studies about learning in students on the autism spectrum, such as those

reviewed by Chiang and Lin (2007). A large portion of these studies focus on K-12

students, and particularly elementary students, but some of the ideas and procedures in

those studies lend themselves to use in a post-secondary context.

While most mathematics-related research on people on the autism spectrum also takes

place among younger children, there have been multiple reports, such as those from

James (2010) and Iuculano et al. (2014) which indicate an association between autism

and heightened mathematical interest and ability. Due to the young age of the students,

however, it remains to be seen whether those tendencies extend to university-level

mathematics content.


I think that Vygotksian educational research provides a promising theoretical

framework for further examination of the results of these comparisons, and possible

reasons behind the observed differences. Vygotsky did not write about autism

specifically (since the diagnosis did not exist at the time). However, Vygotsky (1993)

did write about defectology (as disability research was called at the time, though it had

a narrower scope than the modern concept), criticizing the focus that others took solely

on the deficiencies that they saw in children. He stated that children had the potential

to develop strengths to “overcompensate” for these deficiencies, in particular when

MEDS-C 2015 Truman

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those efforts were targeted toward a particular goal. He criticized other educational

methods at the time which kept people with disabilities out of mainstream society,

giving a sparse curriculum and low expectations for their performance in the school or

later in life; many of his criticisms and more still apply to some programs today. Since

Vygotsky's theory of development heavily involves cultural artifacts, and as he points

out those artifacts developed for the 'typical' child, development will differ not just

based on the initial differences in the child, but on how those differences affect the use

of cultural tools in development. This is strongly related to current social

constructionist models of disability, as outlined by Rodina (2006), as well as the social

model of disability seen in contemporary activism.

The Vygotskian conception of inner speech also has interesting applications for

autism-related research. According to Vygotsky (1962), speech develops from a single

general function to distinct functions for oneself and for others. The 'for oneself'

development is first into egocentric speech (still out loud); while this refers to the same

empirical phenomenon as Piaget's egocentric speech, the conception of development

here is different. Instead of simply ending, Vygotsky's egocentric speech develops into

inner speech, which is no longer vocalized. However, the differences are more

extensive; from Vygotsky's study of egocentric speech, he found that it gradually

develops into a form which has greater distinction from social speech, more

idiosyncratic and less understandable to others, before it ultimately develops into inner

speech. He concluded that this form of speech relies heavily on internal conceptions

and senses which could not be used in social speech, and thus it is something more than

just speech without vocalization.

AUTISM-RELATED RESEARCH IN THE VYGOTSKIAN FRAMEWORK

Some research studies have been conducted attempting to determine the level of

development of inner speech in autistic children, with varying results. One study

conducted by Wallace, Silvers, Martin, and Kenworthy (2009) indicated that

“individuals with autism do not effectively use inner speech during the completion of

cognitive tasks”, while another study from Williams, Happé, and Jarrold (2008)

indicated that “individuals with ASD use inner speech to the same extent as individuals

without ASD of a comparable mental age.” A third study conducted by Russell-Smith,

Comerford, Maybery, & Whitehouse (2014) examined this in the particular context of

executive function-related tasks (they appear to implicitly equate this to Vygotsky's

'self-regulation', and one of their citations is the Wallace et al. paper which does this

explicitly), and involved a greater variety of conditions to alter the use of inner speech.

They also found discrepancies in inner speech use in the same direction as Wallace et

al., as well as a correlation between verbal test scores and task performance in the ASD

participants. However, while the Wallace et al. study found a statistically significant

worse performance in the ASD group in the control condition, the Russell-Smith et al.

study did not.

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It should be noted that all of these studies were mostly done on children, and only one

had any participants over 18, so they were not aimed toward the same population as my

own research. Also, in all three studies (like in most autism-related studies), the vast

majority of study participants were boys. In this case, there is not a sufficient sample to

determine if the outcomes would be different with girls (there is a widely reported

disparity in diagnosis, but not necessarily in occurrence). Another unexpected finding

was that of the three studies, the one where the ASD study participants had a DSM-IV

diagnosis of Asperger syndrome is the Wallace study, which found a difference in both

inner speech use and general performance. Given that the main difference between the

autism and Asperger diagnoses was in verbal criteria, this runs counter to expectation

based on those diagnostics, and provides more evidence that the developmental paths

here may be more complex than these studies are able to discern.

Some reports, such as one by Patterson (2008), show cases of people on the autism

spectrum who do not have much social interaction and often read a variety of 'higher

level' books. Typically, they are noted for having an extensive vocabulary, but not the

skills to relate to typically developing children their own age. While in Vygotsky's

model, speech is most often mediated primarily by direct interaction with others, his

defectology work also suggested that alternate means could be used to work toward the

same goals. In this case, it seems plausible that the children involved here may have

had their reading as a large component of mediation for speech development, and thus

their 'unusual' vocabulary would appear to be a natural consequence of this alternative

mediation. It is also worth noting that often extensive reading or other interests in

children on the autism spectrum are identified in the category of “special interests”,

and while some methods of therapy try to diminish those interests, Vygotsky's ideas in

his defectology writing about using the child's strengths fit more with the idea of using

them as part of the educational process.

Since inner speech is by its nature not directed toward others, it is reasonable to

consider a process of formation could happen which does not heavily involve direct

interaction with others (though cultural knowledge is still mediated from the authors

through their works). Also, particularly with nonfiction, we would not expect these

books to mediate elements of social interaction that are not described in their pages,

giving less progress in that area. The discrepancies we have seen in studies of inner

speech and task performance among children on the autism spectrum could be a result

of the kinds of mediation that are available to the participants and how those work with

the particular variations of the child in question (since there is a wide range of variation

in the autism spectrum). Also, while Vygotsky notes that “the potential for expressive

intonation, mimic, and gesture, is absent in written speech,” since it is precisely those

things that are often most troublesome for autistic speakers, that could be another

reason for them to be stronger with written than oral speech, and this could play into

development.

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INTERVIEW PROCEDURES

I started out with the intent to find students who were on the autism spectrum and

currently taking one or more mathematics courses at SFU. My advisor and I requested

that the Centre for Students with Disabilities at SFU help us recruit for students who

would be qualified for the study, and were put in touch with one student who was

willing and able to do the interviews. It should be noted that this method will only

identify those students who are both able to seek assistance from such a center and see

the need to do so, and this constitutes only a portion of the fairly wide autism spectrum.

The interviews were conducted with audio recording, and portions of the interview

transcribed when they were examined.

The student I interviewed, Joshua (a pseudonym) was studying integral calculus and

linear algebra. I conducted interviews every week for this term. These were scheduled

for one hour, but were sometimes continued for a short time past the scheduled hour. I

typically started by asking the student to share any particular thoughts on the week's

course materials. I also asked various questions and assigned tasks related to the

covered course material. Some of these were tasks that have been used with typical

student populations in the literature, such as the example-generation tasks used by

Bogomolny (2006) and the Magic Carpet Ride sequence used by Wawro, Rasmussen,

Zandieh, Sweeney, & Larson (2012). I have also given other mathematical tasks not

directly related to the material covered in the courses being taken, such as the

paradoxes examined by Mamolo and Zazkis (2008); one reason for this was the

interplay between visual and algebraic explanations seen in some student responses to

these paradoxes. For the most part, I attempted to examine the student's thoughts

without interference, but I would try to explain anything which was still incorrect or

unresolved at the end of the session.

There were several reported characteristics of people on the autism spectrum which I

thought could be promising for mathematics education research. In particular, I was

interested in details of prototype formation, special interests, and geometric

approaches. I will detail each of these with a comparison to the particular findings

relevant to them in Joshua's case. One smaller finding here was that Joshua reported a

heavy use of the textbook and not much gained from the lecture; this is opposite to my

observations of other students and fits in with the possibilities of alternative mediations

of inner speech.

PROTOTYPE FORMATION

I started looking into prototype formation after reading a study by Klinger and Dawson

(2001). It suggested that people on the autism spectrum did not form prototypes of

objects when given tasks asking about group membership, instead taking an approach

based on lists of rules. Although this is presented as a problem, like many other

autism-related studies, I suspected that this approach could be helpful for more abstract

or proof-based mathematics. In Vygotskian terms, it could be regarded as an attempt to

form a scientific concept rather than building it up from an everyday concept. I have

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found many other students having trouble with mathematical questions that appear to

result from a prototype-based approach, and this is particularly true when the course

focuses on mathematical proof. In fact, I found a very similar division reported in

mathematics education research by Edwards and Ward (2004), phrased as lexical or

extracted definitions versus stipulative definitions. This did not appear to be the case

for Joshua; he reported having this kind of thinking in the past, but was quite focused

on “big picture” ideas today (this was, in fact, a recurring phrase in the interviews).

SPECIAL INTERESTS AND LEARNING

I found the idea of 'special interests' (variously known as circumscribed interests,

splinter skills, savant skills, and a variety of other names with varying connotations

within the autistic and research communities) to be applicable to some of my findings.

These are intense, focused interests occurring in people on the autism spectrum (often

studied in children, like much autism-related research). Some of these are

mathematically related, such as a focus on particular facets of arithmetic, prime

numbers, or aspects of geometric shapes, as explored by Klin, Danovitch, Mars, &

Voltmar (2007). The development of these interests in later life is something that does

not appear to have been studied much in previous research, however. Unfortunately,

research in this area (particularly from an Applied Behavioral Analysis framework)

can discount these skills or even view them as detrimental to learning, as seen in

Dawson, Mottron, and Gernsbacher (2008), and may even attempt to eliminate these

skills. From what I have seen, however, such interests can be very helpful in imparting

motivation when viewed from the right perspective. Joshua has reported a strong

interest in chemistry, which he often used in analogies for mathematical concepts in

our interviews. There have been several instances where he has reported more

enthusiasm, better understanding, and better performance when able to see chemical

applications to the topics in the courses, and has sought out additional information

outside the course materials in order to make these connections.

GEOMETRIC FOCUS AND VISUALIZATION

Particularly due to the work of Temple Grandin, one of the most famous people on the

autism spectrum, there is often an association between the spectrum and visualization

or spatial reasoning (Grandin, Peterson, and Shaw, 1998). There could also be a

relationship between this and differences in the mediation of inner speech. While I

would caution against being too broad with an association like that, I did find a strong

preference for visual, spatial, or geometric reasoning in the interviews I conducted.

This was particularly successful with integral calculus, where the student

independently thought about what three-dimensional integrals might be like. The

correct conclusion was reported for a 'flat' extension (of multiplying by a constant

length), and the student did realize that this would not work for more complex

three-dimensional shapes (although not to the point of developing multiple integrals).

Reports of classroom progress continually reflected higher performance in areas that

could be viewed in a geometric or otherwise physical way. Comparing topics across

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the two courses, this leads to some surprising results, such as reported satisfaction with

washer and shell rotations, but issues with algebraic formulas like dot products.

I found the solution Joshua gave in one interview for the first Magic Carpet Ride task

to be particularly notable. I showed the problem setup from the paper by Wawro et al.

(2012), giving the two modes of travel with vectors (3,1) and (1,2), and asking for a

way to get to the house at (107,64). Since I asked this relatively late, Joshua had

already seen the vector material that shows the 'standard' way to do this. However, the

solution he gave was instead done by drawing the vectors out on paper. He measured

(30,10) and (10,20), plotted (107,64), shifted one of the vectors so that it would end at

(107,64), and extended the vectors in order to find via measurement their point of

intersection. (I checked this, and it was accurate enough to provide the correct

solution.) When asked for an algebraic solution, he calculated the equations of the

lines corresponding to those vectors and found their point of intersection, still clearly

based on the same visualization.

PARADOXES

I have also presented several paradox tasks during my interviews. I gave the Hilbert

Hotel and ping-pong ball paradoxes from the paper by Mamolo and Zazkis (2008), as

well as the Painter's Paradox (involving the volume and area of Gabriel's Horn

obtained by rotating 1/x) and an 'infinitesimal staircase' paradox. The ping-pong ball

paradox involves an infinite set of ping-pong balls (numbered 1, 2, 3, …) being

inserted into and removed from a barrel over one minute. In the first 30 seconds, the

first 10 balls are inserted, and the '1' ball is removed. In the next 15 seconds, 11

through 20 are inserted and the '2' ball is removed, and so on. The respondents are then

asked how many ping-pong balls remain in the barrel at the end of the minute. The last

one involves a staircase being divided up into finer and finer steps (at each stage having

perimeter 2) to approach an incline (of length √2). Like many of the students in

previous studies, the student I interviewed found these to be strange and paradoxical.

However, they also appeared to inspire a great deal of enthusiasm; through having a

series of interviews, I was able to see that these paradoxes had inspired independent

thinking outside of the interview sessions. The response provided to the ping-pong

ball paradox was something I also found particularly notable. Joshua first said that

there should be infinitely many, then decided that there should be none after being

asked what the numbers of the remaining balls were. However, the analogy he

provided here was of slowing down molecules at extremely low temperatures in order

to study them; it appeared as if this mental image was being used to accommodate the

presence of the balls at times prior to the conclusion of the experiment. I also found it

notable that I did not see any tendency toward rejecting the mathematical facts after

they had been presented, unlike in many of the students in the prior studies. It is

possible that differences in the formation of the mathematical concepts involved led to

a difference in what parts of the concept are regarded as the most fundamental, so that

the parts considered fundamental here may not be the ones in conflict.

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LIMITATIONS AND SUGGESTIONS

So far, I have only been able to conduct interviews with one research subject, and thus

it is important to be careful not to overly generalize the results seen in the interviews.

Even with more participants, there are some forms of autism which would indicate

notable unwillingness or difficulty in participating in the study, so a study like this can

only target a subset of the population. There are also considerations about the content

that would apply to the courses that Joshua took, although I think that linear algebra is

a fortunate course to have an interviewee from. Some questions that have occurred to

me: Would people on the spectrum with less clearly related special interests still use

them for mathematical analogies? Is there a tendency for either (more specifically) a

bias toward geometric processing or (more generally) a tendency to strongly prefer one

type of processing? Are people on the spectrum generally more inclined to accept

conclusions that are viewed as paradoxical?

References

Autistic Self Advocacy Network (2014). About autism. Retrieved from

http://autisticadvocacy.org/about-autism/

Bogomolny, M. (2006). The role of example-generation tasks in students' understanding of

linear algebra (Doctoral dissertation, Faculty of Education-Simon Fraser University).

Chiang, H. M., & Lin, Y. H. (2007). Mathematical ability of students with Asperger

syndrome and high-functioning autism A review of literature. Autism, 11(6), 547-556.

Dawson, M., Mottron, L., & Gernsbacher, M. A. (2008). Learning in autism. In J. H. Byrne

(Ed.), Learning and memory: A comprehensive reference (Vol. 2, pp. 759-772). Oxford,

UK: Elsevier.

Edwards, B. S., & Ward, M. B. (2004). Surprises from mathematics education research:

Student (mis) use of mathematical definitions. American Mathematical Monthly, 411-424.

Grandin, T., Peterson, M., & Shaw, G. L. (1998). Spatial-temporal versus language-analytic

reasoning: the role of music training. Arts Education Policy Review, 99(6), 11-14.

Iuculano, T., Rosenberg-Lee, M., Supekar, K., Lynch, C. J., Khouzam, A., Phillips, J., . . .

Menon, V. (2014). Brain organization underlying superior mathematical abilities in

children with autism. Biological Psychiatry, 75(3), 223-230.

James, I. (2010). Autism and mathematical talent. The Mathematical Intelligencer, 32(1),

56-58.

Klin, A., Danovitch, J. H., Merz, A. B., & Volkmar, F. R. (2007). Circumscribed interests in

higher functioning individuals with autism spectrum disorders: An exploratory study.

Research and Practice for Persons with Severe Disabilities, 32(2), 89-100.

Klinger, L. G., & Dawson, G. (2001). Prototype formation in autism. Development and

Psychopathology, 13(01), 111-124.

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Mamolo, A., & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in

Mathematics Education, 10(2), 167-182.

Patterson, M. (2008, March 28). My 'little professor'. Chronicle of Higher Education.

Retrieved from http://chronicle.com/article/My-Little-Professor-/14212

Rodina, K. (2006). Vygotsky's social constructionist view on disability: A methodology for

inclusive education. Enabling Lifelong Learning in Education, Training and Development:

European Learning Styles Information Network (ELSIN), University of Oslo: Oslo,

CD/ISDN, 82-8075.

Russell-Smith, S. N., Comerford, B. J., Maybery, M. T., & Whitehouse, A. J. (2014). Brief

report: Further evidence for a link between inner speech limitations and executive function

in high-functioning children with autism spectrum disorders. Journal of Autism and

Developmental Disorders, 44(5), 1236-1243.

Vygotsky, L.S. (1962). Thinking and Speaking. (E. Hanfmann, G. Vakar, & N. Minnick,

Trans.). MIT Press. Retrieved from https://www.marxists.org/archive/vygotsky/works/words/

Vygotsky, L.S. (1929/1993). The Fundamental Problems of Defectology. Plenum Press.

Retrieved from https://www.marxists.org/archive/vygotsky/works/1929/defectology/index.htm

Wallace, G. L., Silvers, J. A., Martin, A., & Kenworthy, L. E. (2009). Brief report: Further

evidence for inner speech deficits in autism spectrum disorders. Journal of Autism and

Developmental Disorders, 39(12), 1735-1739.

Wawro, M., Rasmussen, C., Zandieh, M., Sweeney, G. F., & Larson, C. (2012). An

inquiry-oriented approach to span and linear independence: The case of the magic carpet

ride sequence. PRIMUS, 22(8), 577-599.

Williams, D., Happé, F., & Jarrold, C. (2008). Intact inner speech use in autism spectrum

disorder: evidence from a short‐term memory task. Journal of Child Psychology and

Psychiatry, 49(1), 51-58.

http://chronicle.com/article/My-Little-Professor-/14212

https://www.marxists.org/archive/vygotsky/works/words/

https://www.marxists.org/archive/vygotsky/works/1929/defectology/index.htm

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OVERVIEW OF RESEARCH ON STUDENTS’ VIEWS OF ORAL

ASSESSMENT IN MATHEMATICS

Milica Videnovic


With the increased emphasis on closed book written examinations, there is a critical

need for implementing the oral assessments in mathematics courses. Based on the

overview of research on students’ views of oral assessment in their mathematics

courses, most of the students had positive views towards its use. In this paper, the

research on oral assessment in mathematics was summarized with a specific focus on

students’ views.

RATIONALE FOR THE RESEARCH

For many years, the assessment in mathematics classroom seems to be strictly based on

closed book written examinations. The USA appears to be dominated by closed book

examinations (Gold, 1999; Nelson, 2011). Iannone and Simpson (2011) noted that the

majority of mathematics students in the UK seem to be assessed predominately based

on high stakes, closed book examinations at the end of almost every module. Joughin

(1998) argued that the structure of the assessments today are either closed and formal,

with little interaction between student and assessor(s), or open, with less structure and

the opportunity for dialogue between student and assessor(s).

We might think that written examinations have always been used as a standard practice

of assessing students’ mathematical understanding, but that has not been the case.

Written examination was the norm in the UK, starting at the beginning of the 20th

century. Prior to the beginning of the 20th century, oral examination was a standard

practice in the UK, which later on failed because of accusations of bias and the

apparent efficiency of written exams (Stray, 2001). The four factors that have been

identified as being crucial in this oral/written shift are: the move from group

socio-moral to individual cognitive assessment in the later 18th century; the

differential difficulty of oral testing in different subjects; the impact of increased

student numbers; and the internal politics of Oxford and Cambridge (Stray, 2001).

Today, there are some countries that still maintain an oral assessment as an important

part of their assessment diet, such as Hungary, Italy and the Czech Republic (Stray,

2001).

There is a perception that an oral assessment may make students more anxious than

other forms of assessment. Hounsell, Falchikov, Hounsell, Klampfleitner, Huxham,

Thompson and Blair (2007) noted that “It is not clear whether oral assessments are

scarier or just more novel” (p. 34). Also, Huxham, Campbell and Westwood (2012)

noted that oral assessment anxiety may be primarily related to its unfamiliarity.

Joughin (2007) noted, in his phenomenographic study of student experiences of oral


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presentations, that greater anxiety about oral compared with written assessments was

associated with a richer conception of the oral task as requiring deeper understanding

and the need to explain to others. On the other hand, Joughin (2007) pointed out that

many students described the oral presentations as being more demanding than the

written assignments, more personal, requiring deeper understanding, and leading to

better learning. One of the advantages of oral assessment, noted by Joughin (1998), is

that the interaction allows for probing knowledge through dialogue that is genuinely

individual which “Gives assessment an inherent unpredictability in which neither party

knows in advance exactly what questions will be asked or what responses will be

made” (p. 371). Therefore, oral assessments might be considered fairer than some

other forms of assessments. It makes plagiarizing very difficult, where students must

explain their own understanding using their own words, and unlike other examinations,

oral assessment with a tutor prevents one small gap in knowledge completely stalling a

solution (Joughin, 1998). Also, Joughin (1998) noted that oral examinations are more

authentic than most types of assessments. He described this authenticity as “The extent

to which assessment replicates the context of professional practice or real life” (p.

371).

When it comes to different types of oral assessment, according to Joughin (2010), they

can be categorized into three forms: presentation on a prepared topic (individual or in

groups); interrogation (covering everything from short-form question-and-answer to

the doctoral viva); and application (where candidates apply their knowledge live in a

simulated situation, e.g. having trainee doctors undertaking live diagnoses with an

actor-patient).

There is a very little literature examining the use of oral assessments. Hounsell,

Falchikov, Hounsell, Klampfleitner, Huxham, Thompson and Blair (2007) noted in

their comprehensive review of the literature on innovative assessment that less than 2%

of the papers address the oral assessments. They reviewed the recent UK literature on

‘innovative assessment’ and of 317 papers considered, only 31 dealt with ‘non-written

assessments.’ Within this category, only 13% addressed the use of oral examinations.

PURPOSE

Considering the fact that the literature on oral assessment in higher education is mostly

dominated by the lecturer’s or tutor’s perspective, not much the learner’s (Joughin,

1998), in this paper, I wanted to present the overview of research on students’ views of

oral assessment in mathematics. The purpose of this paper is to share results from

research on students’ views of oral assessment in their mathematics courses, with a

particular emphasis on affective factors such as beliefs, attitudes, emotions, etc. The

need for the study is obvious: oral assessments have a positive impact on students’

learning of mathematics (Boedigheimer, Ghrist, Peterson & Kallemyn, 2015; Fan &

Yeo, 2007; Iannone & Simpson, 2012; Iannone & Simpson, 2014; Nelson, 2011; Nor

& Shahrill, 2014; Odafe, 2006).


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METHODS

There were several criteria used to arrive at the final pool of articles for analysis. The

article had to include empirical-based work and have students enrolled in mathematics

courses as participants, regardless of their age, course level, or level of learning. The

research had to focus on students’ views only. The main electronic database explored

was ERIC, using combinations of key words: oral assessment, oral examination, oral

exam, and mathematics. The electronic database search yielded a total of 26

publications. Abstracts were reviewed to see if the articles met the criteria for inclusion

in this project, and the results of this search gleaned a total of 4 articles. After a review

of abstracts and in-depth analysis of each paper, I was able to find 3 more articles that

appeared to meet the criteria, so the final pool was comprised of a total of 7 articles.

Participants

21 college algebra students (Odafe, 2006)

~ 200 first year, secondary students (Fan & Yeo, 2007)

36 Calculus I students (treatment group)/62% were considered at risk of failing

Calculus I because their placement scores were less than 18/30

96 – 140 Calculus I students (control group)/16% were “at risk” (Nelson, 2011)

- The treatment consisted of voluntary oral assessments offered before every

written examination.

108, first-year, discrete mathematics students, graph theory (Iannone &

Simpson, 2012; Iannone & Simpson, 2014)

28 trigonometry students, 15-16 years old (Nor & Shahrill, 2014)

Representative comments were collected from anonymous end-of-semester

student mathematics course critiques (Boedigheimer, Ghrist, Peterson &

Kallemyn, 2015)

Oral Examination

The structure of the oral examination was the following:

Exam sessions were scheduled ahead of time

Students were given questions before an exam (usually one week before an

exam)

Exams were conducted individually or in groups

Each student would randomly pick a question previously assigned

Exam questions were created to generate discussion among the students

Exams consisted of both presentation and dialogue, had a relatively open

structure, and combined oral medium with writing on a board

The examiner would ask follow-up questions after the student’s response to the

initial questions.


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Tools

There were mainly two instruments designed for data collection, with a purpose to see

students’ perceptions about their own ability to perform, beliefs in the usefulness, and

acceptance of oral assessment/presentation tasks. The two instruments were:

Students’ questionnaire surveys (Boedigheimer, Ghrist, Peterson &

Kallemyn, 2015; Fan & Yeo, 2007; Iannone & Simpson, 2012; Nelson,

2011; Nor & Shahrill, 2014; Odafe, 2006)

Interview with students (Iannone & Simpson, 2014) (19 students agreed to

be interviewed)

RESULTS: ANXIETY, ATTITUDES, AND BELIEFS

Anxiety

Some students reported being anxious (Boedigheimer, Ghrist, Peterson & Kallemyn,

2015; Iannone & Simpson, 2014; Nor & Shahrill, 2014; Odafe, 2006). When it comes

to students’ anxiety, the most common students’ comments were related to lack of

familiarity (Iannone & Simpson, 2014; Nor & Shahrill, 2014). When students were

introduced to the oral assessment, most of them were unfamiliar with this form of

assessment, so it caused them to feel anxious about it.

Attitudes

Most of the students had positive attitudes towards the use of oral assessment (Iannone

& Simpson, 2012; Nor & Shahrill, 2014; Odafe, 2006). Moreover, 67 % of students

preferred the oral to the written examinations, while only 9 % of students favored both

forms of examination equally (Odafe, 2006). Of 85, 73 students commented, and most

of them were being positive about oral assessment as an alternative to weekly

coursework sheets, but negative about them as an alternative to examinations (Iannone

& Simpson, 2012). Also, 65.4% of the students agreed that they ‘like to do

Mathematics poster and oral presentations during Mathematics lesson’ in comparing

with only 34.6% who disagreed (Nor & Shahrill, 2014).

Beliefs

Taken from the provided research, students believed that oral assessments in

mathematics are:

Reactive to students’ needs (Boedigheimer, Ghrist, Peterson & Kallemyn, 2015;

Iannone & Simpson, 2012; Iannone & Simpson, 2014; Odafe, 2006)

Promote deep comprehension of the learned material (Fan & Yeo, 2007;

Iannone & Simpson, 2012; Iannone & Simpson, 2014; Nelson, 2011; Nor &

Shahrill, 2014; Odafe, 2006)

Encourage students to deeply/actively engage with the course material

(Boedigheimer, Ghrist, Peterson & Kallemyn, 2015; Iannone & Simpson, 2012;

Nor & Shahrill, 2014; Odafe, 2006)


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Reactive to students’ needs

When it comes to reactiveness of oral assessments to students’ needs, 67 % of students

claimed that they studied and learned more in the process, and that oral assessment

enabled them to identify their mistakes and misconceptions and got them corrected

immediately, by being provided with immediate verbal feedback during the

examination (Odafe, 2006). Moreover, the assessor could respond to the performance

of the student and to their needs, and this brought up the opportunity for the assessor to

help the student over a gap in knowledge (Iannone & Simpson, 2014).

Promote deep comprehension of the learned material

In one study, 45% of the students believed that ‘doing mathematics oral presentation

helps me to be more aware of my understanding of mathematics,’ vs. 32.5% who

disagreed and 22.5% who neither disagreed nor agreed (Fan & Yeo, 2007). Also,

42.5% of the students believed that ‘doing mathematics oral presentation makes me

think broader and deeper about mathematics,’ vs. 35% who disagreed and 22.5% who

neither disagreed nor agreed (Fan & Yeo, 2007). More than 70% of the students

believed that ‘listening to other classmates’ oral presentation is helpful for me in

learning mathematics’ (Fan & Yeo, 2007). Some of the students’ comments, related to

the use of oral assessment, were such as:

“Let me test my knowledge, and allowed me to recognize my gaps of understanding in

some areas” (Nelson, 2011, p. 51).

“We went over problems in depth: it helped work out any things we didn’t understand”

(Nelson, 2011, p. 51).

Furthermore, the treatment students, who were considered at-risk, did significantly

better than the control group on course grades, completed Calculus I and enrolled in

and passed Calculus II at dramatically higher rates than at-risk students in the control

group (Nelson, 2011). Many of these students reported that oral assessments were a

major reason for their success. Also, 88.5% of the students believed that doing

mathematics poster and oral presentations helped them to learn mathematics in

comparing with only 11.5% who disagreed (Nor & Shahrill, 2014). The mean of the

pre-test of the whole class were found to have improved from 29.80 to 67.05 in the

post-test, which was an increase by 37.25 (Nor & Shahrill, 2014). These students were

given pre-test prior to the 'project' (poster and oral presentations) and a post-test

after. The questions on the pre-test and post-test were identical. On the other hand,

some students saw the oral assessment as the potential value for promoting

understanding of the material, while not replacing the final, high stakes, written closed

book exam (Iannone & Simpson, 2012).

Encourage students to deeply/actively engage with the course material

Of 21 college algebra students, 19 admitted putting in more time and effort into

preparing for the oral examination, compared to the time put into preparing for the

traditional pen and paper examination. The other 2 students claimed to have spent just


127

about the same amount of time and effort on both forms of assessment (Odafe, 2006).

The oral assessments made students think more about the material, encouraged them to

understand the material better, made them remember the material longer after the

assessment, and did not allow them to get high marks without understanding (Iannone

& Simpson, 2012).

CONCLUSIONS

At the end, we can say with certainty that oral assessment can be a useful tool for

assessing the students’ conceptual understanding of the learned material and result in

increased students’ learning of mathematics (Boedigheimer, Ghrist, Peterson &

Kallemyn, 2015; Fan & Yeo, 2007; Iannone & Simpson, 2012; Iannone & Simpson,

2014; Nelson, 2011; Nor & Shahrill, 2014; Odafe, 2006). During the oral examination,

teachers can discover gaps in student knowledge or common misconceptions that can

be corrected immediately. Moreover, it provides teachers with the opportunity to

actively listen to students communicate mathematics and value the students'

contributions. Therefore, when students realize that their contributions have been

valued, they will start to appreciate the subject of mathematics more, and this could

change completely their attitudes towards the mathematics. In addition, if we, as we

do, acknowledge that each student learns differently, then having a common approach

to assessment would be inadequate. Hence, educators accept the need for differentiated

instruction, so in order to deal with the individuality and variability of students, they

also need to accept the need for differentiated assessment to represent the learning of

the fractured student collective (Liljedahl, 2010).

Future Research

Based on these findings on students’ views of oral assessment in mathematics, the

following questions can be considered for future research:

What kind of mathematical understanding can students show during the oral

assessment?

What kind of students’ mathematical understanding can examiner discover

during the oral assessment?

Do the students’ responses in oral and written examinations differ between

different types of questions?

References

Boedigheimer, R., Ghrist, M., Peterson, D., & Kallemyn, B. (2015). Individual Oral Exams in

Mathematics Courses: 10 Years of Experience at the Air Force Academy. PRIMUS, 25(2),

99-120.

Fan, L., & Yeo, S. M. (2007). Integrating oral presentation into mathematics teaching and

learning: An exploratory study with Singapore secondary students. The Montana

Mathematics Enthusiast, Monograph, 3, 81-98.


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